Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra
Simon Becker Federica Gregorio Delio Mugnolo
math.SP math-ph math.AP math.MP



We analyze properties of semigroups generated by Schrödinger operators -Δ+V or polyharmonic operators -(-Δ)^m, on metric graphs both on L^p-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kreĭn-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.






RAFT: Recurrent All-Pairs Field Transforms for Optical Flow
    Zachary Teed Jia Deng
    December 30, 2023
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§ INTRODUCTION



Differential operators on so-called metric graphs have been introduced in the mathematical literature by Lumer in <cit.>, cf. <cit.> for references to earlier investigations in the applied sciences.
Most of ongoing research is devoted to the case of finite graphs, i.e., of metric graphs whose underlying discrete structure consists of finitely many vertices and edges.

Restricting to the case of finite graphs largely simplified earlier investigations: it is nowadays known 

 that the theory of Laplace-type operators becomes notably subtler in the infinite case, as recently shown by e.g. <cit.> for the case of difference operators on combinatorial graphs and <cit.> for Schrödinger operators on metric graph. The first and most obvious side effect is that on general infinite graphs the compactness of the resolvent may be lost and the spectral properties of the Laplacian on different L^p-spaces and Banach spaces of continuous functions may no longer coincide. Most importantly, one may have to take care of boundary growing at infinity and, in turn, of non-uniqueness of solutions of diffusion equations.



However, most existing results heavily rely on Hilbert space machinery; so far, to the best of our knowledge not much attention has been devoted to the case of linear differential operators acting on spaces of continuous functions over an infinite metric graph 𝒢. For results on finite graphs, see <cit.>. The case of infinite graphs that we address in this paper is subtler, as spaces C(𝒢) are no longer contained in L^2(𝒢).
To mention just one important difference: Unlike in the case of the Hilbert space L^2(𝒢) there is in general no strongly continuous heat semigroup acting on C(𝒢),  as already the trivial case of 𝒢=ℝ shows. 
However, among other things we are going to show in this article that both the space of continuous functions vanishing at infinity and the space of bounded uniformly continuous functions do yield strongly continuous heat semigroups.

Indeed, we are going to show that polyharmonic operators and a broad class of Schrödinger operators drive well-posed parabolic equations on suitable spaces of continuous functions acting on general, possibly non-equilateral infinite graphs, as well as on more usual L^p-spaces. Our way to prove this is unusual and, we believe, among the main points of interest of our article: instead of checking that the assumptions of Hille–Yosida-type results are satisfied, we are able to provide a semi-explicit formula for the integral kernel of the relevant semigroups, in the case of spatially constant potentials. We do so by developing a convolution calculus for functions supported on graphs and showing that semigroup kernels on  extend to semigroup kernels on infinite graphs under rather mild summability assumptions.
An analogous (semi-)explicit formula for the solution to the heat equation on finite metric graphs has been found by Roth in <cit.> and then extended by Cattaneo to infinite equilateral trees and finally to general equilateral infinite graphs in <cit.>, respectively. 

The fundamental object of our theory is the so-called Kirchhoff Laplacian Δ, i.e., the operator acting as second derivative on the space of edgewise smooth functions that satisfy continuity condition across each vertex while their normal derivatives sum up to zero about each vertex (Kirchhoff condition). We refer to Section <ref> for a precise definition.


The explicit construction of semigroups generated by Schrödinger and polyharmonic operators is stated in Theorem <ref> and presented in Section <ref>. We then show in Theorem <ref> that the above semigroups are also ultracontractive from which we conclude that the spectrum of Schrödinger operators is independent of p ∈ [1,∞) and coincides with the spectrum on C_0(𝒢) and BUC(𝒢), cf. Theorem <ref> and Corollary <ref>. 

In this article, we connect this explicit construction with the L^p-invariance properties of the spectrum of the generator investigated by Voigt and his co-authors in <cit.>. We extend the results for Schrödinger semigroups on ^d to metric graphs and are able to conclude essentially from the spectral invariance a variety of semigroup properties – including contractivity (Proposition <ref>), positivity (Proposition <ref>), analyticity (Theorem <ref>), the asymptotic behavior (Corollary <ref>) and the (strong) Feller property (Corollary <ref>) – for the semigroup. 


A reduction principle connecting the eigenvalues of the Kirchhoff Laplacian on a finite metric graph to those of the normalized discrete Laplacian on the underlying combinatorial graph was first observed by von Below in <cit.>. His results were extended to the different parts of the spectrum of more general operators on possibly infinite graphs, beginning with <cit.>; we refer to <cit.> and references therein for later refinements of Cattaneo's results. In Section <ref> we prove that this spectral correspondence between combinatorial and metric graphs carries over to general Schrödinger operators and can be extended to general L^p-spaces and certain spaces of continuous functions on the one hand and specific discrete spaces. These results are summarized in Theorem <ref>.

In Section <ref> we then show that we are also able, at least in the case of the Laplacian on equilateral metric graph, to identify the continuous spectrum with the continuous spectrum of the discrete Laplacian on the vertices. 

It should be mentioned that Exner, Kostenko, Malamud, and their co-authors have proved interesting connections between the Laplacian with δ-interactions on possibly non-equilateral metric graphs and a certain generalized discrete Laplacian on the underlying combinatorial graph <cit.> – remarkably, not only at a spectral level, but also concerning parabolic properties: for instance, ultracontractivity of the semigroup generated by the former operator is equivalent to ultracontractivity of the semigroup generated by the latter, with equal dimension.

Sections <ref> and <ref> contain applications of our results. In Section <ref> we construct a Feller process on metric graphs associated with the Laplacian as its generator, which can be interpreted as a Brownian motion on metric graphs, thus extending the construction carried out on finite graphs in <cit.>. We also derive a Feynman–Kac formula in Prop. <ref> and obtain pointwise heat kernel estimates in Theorem <ref>. In Section <ref>, we study the properties of a parabolic Anderson model (parabolic equation with random Schrödinger operator as its generator) on metric graphs.

Outline of article


  * Definition of Kirchhoff-Laplacian, Sec.  <ref>.

  * Explicit construction of heat semigroup, Sec.  <ref>.

  * Spectral independence and Schrödinger semigroups, Sec.  <ref>.

  * Properties of Schrödinger semigroups, Sec.  <ref>.

  * Point spectrum of Schrödinger operators, Sec.  <ref>.

  * Continuous spectrum of the Kirchhoff-Laplacian, Sec.  <ref>.

  * Markov semigroup and Brownian motion on metric graphs, Sec. <ref>.

  * Parabolic Anderson model on metric graphs, Sec. <ref>.


Acknowledgments
The authors wish to thank Sebastian Mildner (TU Dresden) for fundamental contributions to this paper.

Assumptions & Notation 

The metric graph
Let 𝒢=(𝒱, ℰ) be a countable oriented connected locally finite (i.e.  the degree of every vertex is finite) graph without self-loops.
Here 𝒱 is the vertex set and ℰ the set of edges. The length of all edges ∈ℰ is constrained uniformly from above and below by constants ℓ_↓ and ℓ_↑

    0<ℓ_↓≤||≤ℓ_↑ < ∞.
 

We define ℰ_ to be the set of edges adjacent to a vertex . Moreover, we assume that 𝒢 is oriented such that every edge  gets assigned an initial i() and terminal vertex t(). The cardinality of a vertex  is defined as d_ := |ℰ_| and we assume 𝒢 to be uniformly locally finite, i.e., d_^*:=sup_∈𝒱 d_<∞.


We can, without loss of generality, identify all edges  with intervals (0,||) such that 𝒢 becomes naturally a metric space. For a graph with edges ∈ℰ we define the union of edges with either orientation, i.e., ± where initial and terminal vertex are interchanged, by 𝒢. For ' ∈ℰ(𝒢) we define :=|' | with ∈ℰ to be the unique edge that either satisfies = ' or =-'.







A path P along m ∈ℕ_0 edges is a finite sequence of edges (_0,...,_m) ⊂ℰ(𝒢)^m+1 such that t(_j)=i(_j+1) for each 0≤ j ≤ m-1. We also define the set of paths along m ∈ℕ_0-many edges between edges  and ' by 𝒫_,'(m). The length of such a path is defined as | P | :=∑_i=0^m-1|_i |. 

We assume that there is a non-zero conductivity c() associated with every edge ∈ℰ(𝒢). For the same edge with opposite orientation, -, we set c(-):=c().
As the conductivity of a vertex , we define c():=∑_∈ℰ_ c() which is the sum of all conductivities of adjacent edges.



When analyzing general graphs of the above type in this article satisfying the preceding assumptions, we sometimes have to impose the following assumption on the growth of the graph and conductivities in parts of Sections <ref> and <ref> to conclude spectral equivalence.

[Growth condition-General graphs]

We assume that there are two constants k_↓,k_↑ such that all conductivities satisfy k_↓≤ c()≤ k_↑. Moreover, the number of edges is supposed to grow only sub-exponentially, i.e. let k(r) be the number of edges contained in a ball or radius r with respect to a fixed reference vertex of the graph, then for all ε>0 we have k(r)/e^ε r = o(1) as r →∞.



Mathematical notations.
B_r(x) is the ball of radius r centered at x.

For p ∈ [1,∞), we consider the L^p-spaces with measure dc:=∏_∈ℰ(𝒢) c()  dλ_ where λ_ is the standard Lebesgue measure on some edge. The dual of L^p is then still L^q with p^-1+q^-1=1 for p ∈ [1,∞) and q ∈ (1,∞] due to the dual pairing

    (u,v)_L^p × L^q = ∫_𝒢 u(x) v(x) dc(x).

Instead of writing dc(x), we will usually just write dx in the sequel. 



Analogously, the discrete ℓ^p-spaces on the vertex set 𝒱(𝒢) are given by

    ℓ^p(𝒱(𝒢)):={ z : 𝒱(𝒢) →ℂ; ‖ z ‖_ℓ^p := ( ∑_∈𝒱(𝒢)| z() |^p c() )^1/p < ∞}

with dual pairing (u,w)_ℓ^p×ℓ^q:=∑_∈𝒱(𝒢)  c() (u() w()).
We write C(𝒢) for the space of continuous functions on the graph (requiring continuity also at the vertices).
The spaces of discrete and continuous functions 

    c_0(𝒱(𝒢)),ℓ^∞(𝒱(𝒢)) and C_0(𝒢),BUC(𝒢), L^∞(𝒢)
 
are defined with the standard (i.e., unweighted) supremum norm using the metric structure of the graph and do not depend on conductivities.

As usual, ℓ^∞ can be identified with the double-dual of c_0 and the dual space of ℓ^1. Moreover, both C_0 and BUC are closed subspaces of L^∞.

Here, C_0 is the space of functions vanishing at infinity and BUC is the space of bounded uniformly continuous functions.

We denote the point spectrum of an operator T by σ_p(T):={λ∈ℂ: ker(T-λ) ≠{0}} and the continuous spectrum by 

    σ_c(T):={λ∈ℂ: T-λ is not bijective, but has dense range and is injective}.


We can view ℰ(𝒢) as the collection of edges (without vertices), where ℰ(𝒢) is naturally identified as a smooth disconnected Riemannian manifold.

The space of distributions 𝒟'(ℰ(𝒢)):=⊕_∈ℰ(𝒢)𝒟'() on ℰ(𝒢) is the space of linear forms on C_c^∞(ℰ(𝒢)) such that for every compact set K ⋐ℰ(𝒢) there are C>0 and k ∈ℕ such that | u(φ) |≤ C ∑_|α|≤ k‖φ^(α)‖_∞ for all φ∈ C_c^∞(K).
In particular, we can define the locally convex space ℰ(ℰ(𝒢)), the set of C^∞(ℰ(𝒢)) functions with the topology defined by seminorms φ↦∑_α≤ ksup_x ∈ K ⋐ℰ(𝒢)|φ^(α)(x) |,
where K ranges over all compact subsets of ℰ(𝒢) and k over all integers.

The dual space ℰ'(ℰ(𝒢)) is the space of compactly supported distributions on ℰ(𝒢) that are of finite order. In fact, take ν∈ℰ'(ℰ(𝒢)), then standard results on locally convex spaces imply that there are C>0, k ∈ℕ and K ⋐ℰ(𝒢) such that |ν(φ) |≤ C ∑_α≤ k sup_x ∈ K ⋐ℰ(𝒢)|φ^(α)(x) |  ∀φ∈ C^∞(ℰ(𝒢)).
It is then obvious that if (φ) ⊂ℰ(𝒢) \ K then ν(φ) = 0, i.e., the support of ν is contained in K and the order of ν does not exceed k.


For m ∈ℤ, we define the Sobolev spaces H^m(𝒢) as the Hilbert space direct sum H^m(𝒢) := ⊕_∈ℰ(𝒢) H^m().
We define the function spaces H^m_comp(𝒢) := ⋃_A ⊂ℰ(𝒢): | A | < ∞ H^m(A) equipped with the inductive limit topology and spaces 

    H^m_loc(𝒢) := { u ∈𝒟'(ℰ(𝒢)); ∀φ∈ C_c^∞(ℰ(𝒢)): φ u ∈ H^m(𝒢) }.

The topology on space H^m_loc(𝒢) is generated by seminorms p_φ(u):= ‖φ u ‖_H^m(𝒢) for φ∈ C_c^∞(ℰ(𝒢)).
We then have the continuous inclusions, if we again consider spaces with their inductive limit topologies, ⋂_m ∈ℤ H^m_loc(𝒢) ↪ℰ(ℰ(𝒢)) and ℰ'(ℰ(𝒢)) ↪⋃_m ∈ℤ H^m_comp(𝒢).









§ DEFINITION OF THE KIRCHHOFF-LAPLACIAN



In this preliminary section, we introduce the central object of this article, the Kirchhoff-Laplacian Δ on various function spaces that we will analyze more thoroughly in this article.

We consider a collection (c())_∈ℰ(𝒢) of conductivities. The conductivities enter in the Kirchhoff boundary conditions[sometimes also called Neumann conditions],

    ∑_∈ℰ_(𝒢) ∂ u_∂
    n_() =0,

 that we impose at every ∈𝒱(𝒢). In this condition, both the orientation of the graph and the conductivities enter 
    ∂ u_∂ n_():=
    lim_h → 0_+ c()(u_(h)-u_(0))/h     =i(), 
    
    
    -_h → 0_+ c()(u_(1-h)-u_(1))/h     =t().

We define the domain of the operator Δ on either X_c=C_0(𝒢), X_c=BUC(𝒢) or X_c=L^p(𝒢) with p ∈ [1,∞) by

    D(Δ_X)   := { u ∈ X_c : Hu ∈ X_c, u ∈ C(𝒢), u  satisfies (<ref>)} and (Δ_Xu)_:=u”_;

that is, at each vertex elements u of D(Δ_X) satisfy so-called natural (or standard) conditions: continuity along the vertices for u and Kirchhoff condition on the normal derivative.
Furthermore, powers Δ_X^m are then iteratively defined on D(Δ_X^m+1) :={ u ∈ D(Δ_X^m); Δ u ∈ D(Δ_X)}. This means that all odd derivatives are required to satisfy a Kirchhoff condition and all even ones a continuity condition at the vertices. 

The operator Δ_L^2 is a self-adjoint operator on L^2 <cit.>. Moreover, for general spaces X as above, the operator Δ_X is closed with the choice of domain (<ref>) as the following Lemma shows: 


The domain D(Δ_X) is dense in X_c and the operator -(-Δ_X)^m is closed on D((-Δ_X)^m) with respect to the graph norm ‖ f ‖_Δ_p:= ‖ (‖ f ‖_L^p, ‖ (-Δ)^m f ‖_L^p ) ‖_ℓ^p with the respective p ∈ [1,∞) for spaces X_c=L^p(𝒢) and p =∞ for X_c=C_0(𝒢) and X_c=BUC(𝒢).


We start by defining Banach spaces X_c^(2)

    C_0^2(𝒢)   :={ (f_)_∈ℰ(𝒢); f_∈ C^2[0,||]; (‖ f_‖_C^2[0,||])_∈ c_0(ℰ(𝒢))}, 
    
    W^2,p(𝒢)   :={ (f_)_∈ℰ(𝒢); f_∈ W^2,p[0,||]; (‖ f_‖_W^2,p[0,||])_∈ℓ^p(ℰ(𝒢))}
    
    BUC^2(𝒢)   :={ (f_)_∈ℰ(𝒢); f_∈ C^2[0,||]; (‖ f_‖_C^2[0,||])_∈ℓ^∞(ℰ(𝒢)),  and 
             ∀ε>0  ∃δ>0  ∀ x,y ∈ (0,||)  ∀∈ℰ(𝒢): 
             | x-y |≤δ⇒| f_(x)-f_(y) | ,| f”_(x)-f”_(y) |≤ε}.


That the above spaces are Banach spaces with respect to the graph norm 

    ‖ f ‖_Δ_p:= ‖ (‖ f ‖_L^p, ‖Δ f ‖_L^p ) ‖_ℓ^p
 
follows easily from the Cauchy–Schwarz and Gagliardo–Nirenberg interpolation inequality 

    ‖ f' ‖^p_L^p   = ∑_e ∈ℰ(𝒢) c() ‖ f' ‖^p_L^p( )≲∑_e ∈ℰ(𝒢) c() (‖ f ‖_L^p( )^p/2‖Δ f ‖_L^p( )^p/2+ ‖ f ‖^p_L^p( )) 
       ≲√(∑_e ∈ℰ(𝒢) c() ‖ f ‖^p_L^p())√(∑_e ∈ℰ(𝒢) c() ‖Δ f ‖^p_L^p()) +∑_e ∈ℰ(𝒢) c() ‖ f ‖^p_L^p()
       =‖ f ‖^p/2_L^p‖Δ f ‖^p/2_L^p +‖ f ‖^p_L^p .

We then introduce functionals F_ for vertices ∈𝒱(𝒢) and G_,' for adjacent edges ,' ∈ℰ_(𝒢) by

    F_: X_c^(2)∋ f   ↦∑_∈ℰ_(𝒢)∂ f_/∂ n_()  and  G_,': X_c^(2)∋ f↦ f_()-f_'().

Thus, let X_c^(2) be any of the spaces in (<ref>), then we can define

    D(Δ_X)   :=X_c^(2)∩⋂_∈𝒱(𝒢)ker(F_) ∩⋂_,' ∈ℰ(𝒢); ∩' ={}ker(G_,').

The operators Δ_X are closed since functionals (<ref>) are continuous as maps from X_c^(2) to . The domain D(Δ_X) is dense in X_c as it contains C_c^∞(𝒢). In the case of X_c=BUC(𝒢) ∋ f, it suffices to subdivide for every ε_n=1/n each edge into intervals [x_i,y_i] of sizes between δ_n/2 and δ_n such that | x-y |≤δ_n implies | f(x)-f(y) |≤ε_n. On each of these intervals, we can then choose a function g interpolating between f(x_i) and f(y_i) such that g'(x_i)=g”(x_i)=g'(y_i)=g”(y_i)=0 and g ∈ D(Δ_BUC(𝒢)).



 





§ CONVOLUTIONS ON METRIC GRAPHS AND EXPLICIT CONSTRUCTION OF HEAT SEMIGROUP




In this section, we develop a general framework to naturally extend to metric graphs a class of convolution semigroups on the real line , with sufficiently rapidly decaying integral kernels, to semigroups on 𝒢, building on ideas from  <cit.>. To this aim, we start by defining transfer coefficients:










[Transfer coefficients]

For two edges ,' ∈ℰ(𝒢) we define the transfer coefficient

    𝕋_,':   =(2 c()c(t())-δ_,-')δ_t(),i(')=
      2 c()/c(t()),    if t()=i('), ' ≠ -,
    
      2 c()/c(t())-1,    if t()=i('), ' =-,  and
    
      0,    otherwise.

For each path P=(_0,..,_m) we denote by 𝕋_P the product of transfer coefficients along that path 𝕋_P:=∏_j=0^m-1𝕋__j,_j+1.






We now turn to the construction of the integral kernel on the graph and consider intervals I_m:= [m ℓ_↓ ,∞) for integers m ∈ℕ_0. Then we can define a weighted L^1-norm 

    ‖ f ‖_ℒ^1 :=∑_m ∈ℕ_0 3^m‖ f ‖_L^1(I_m)

on the Banach space ℒ^1 of even functions for which the above sum is finite 

    ℒ^1:={ f ∈ L^1(ℝ); f(-∙)=f(∙)  and ‖ f ‖_ℒ^1<∞}.




We consider exponential scaling in the definition of ℒ^1 due to the –in general– exponential scaling in the number of paths a graph admits. This, we illustrate with the following simple example: 
[Paths on ]
We consider a graph 𝒢 with edges {(n,n+1);n ∈} and vertices 𝒱(𝒢)=.


  * If the conductivities are constant c_(n,n+1)=c_(n+1,n+2) for all n ∈, 
the transfer coefficients are 𝕋_(n,n+1),(n+1,n+2)=1 and 𝕋_(n,n+1),(n,n+1)=0 since there is full transmission and no back-scattering at the vertices. This implies that there is only a single path from the edge := (n,n+1) to ':=(n+k,n+(k+1)). 

  * If the conductivities are not all the same then there are in general paths of length 2m+k from  to ' for any integer m ≥ 0. Thus, the total number of such paths of length 2m+k is 2m+km and scales exponentially in m (Stirling's formula).




The Banach space ℒ^1 is a Banach algebra with respect to the standard convolution on .


We have from an immediate computation with ϕ_n:=‖ f ‖_L^1[n ℓ_↓,(n+1) ℓ_↓] and γ_n:=‖ g ‖_L^1[n ℓ_↓,(n+1) ℓ_↓]

    ∫_m ℓ_↓^(m+1) ℓ_↓| (f*g)(t) | dt ≤   ∫_m ℓ_↓^(m+1) ℓ_↓∑_n ∈∫_n ℓ_↓^(n+1) ℓ_↓| f(s) || g(t-s) | ds  dt 
       ≤∑_n ∈ϕ_n(γ_m-(n+1) + γ_m-n).


This implies that 

    ‖ f*g ‖_ℒ^1   ≤∑_m∈_0,n∈ 3^ m ϕ_nγ_m-(n+1) + ∑_m∈_0,n∈ 3^ m ϕ_nγ_m-n
       ≤ 3 ∑_m∈_0,n∈ 3^| n |ϕ_n 3^| m-(n+1) |γ_m-(n+1) + ∑_m∈_0,n∈ 3^| n |ϕ_n 3^| m-n |γ_m-n
       ≤ 6 ‖ f ‖_ℒ^1‖ g ‖_ℒ^1 + 2‖ f ‖_ℒ^1‖ g ‖_ℒ^1 = 2(3+1) ‖ f ‖_ℒ^1‖ g ‖_ℒ^1.

This concludes the proof.




In the following, we parametrize points of the metric graph 𝒢 by tuples (,ξ) where ∈ℰ(𝒢) and ξ∈ [0,||], with endpoints suitably identified to mirror the graph's connectivity, see <cit.>. For P ∈𝒫_,'(m) with P = (_0,..,_m) and both _0= and _m=', we then have that c(_0)^-1𝕋_P = c(_m)^-1𝕋_-P where -P=(-_m,-_m-1,...,-_0) is the inverted path. 

We can now extend integral kernels f ∈ℒ^1 -of convolution type- on  to integral kernels K_f on products of graphs 𝒢: 

[Integral kernel on 𝒢]
For functions f ∈ℒ^1 we then define an integral kernel K_f on 𝒢×𝒢 by the sum K_f :=K^(1)_f+ K^(2)_f, where K^(1)_f:=K^(1a)_f+K_f^(1b) and K^(2)_f:=K^(2a)_f+K_f^(2b) are defined by

    K^(1a)_f((,ξ),(',ξ'))   :=c( )^-1( δ_, ' f(ξ'-ξ)+∑_m ∈ℕ∑_P ∈𝒫_,'(m)𝕋_P f(ξ'+| P |-ξ) ) 
    
    K^(1b)_f((,ξ),(',ξ'))   :=c( )^-1∑_m ∈ℕ∑_P ∈𝒫_,-'(m)𝕋_P f(|' |-ξ'+| P |-ξ) 
    
    K^(2a)_f((,ξ),(',ξ'))   :=c( )^-1∑_m ∈ℕ∑_P ∈𝒫_-,'(m)𝕋_P f(ξ'+| P |-(||-ξ)) 
    
    K^(2b)_f((,ξ),(',ξ'))   :=c( )^-1∑_m ∈ℕ∑_P ∈𝒫_-,-'(m)𝕋_P f(|' |-ξ'+| P |-(||-ξ))

where | P | is the length of path P.


The four sums above appear by considering paths between points (,ξ),(',ξ'), where each point on the graph can be represented using either possible orientation of edges ±, ±' on the graph .

Let us first note that the total kernel K_f we defined is symmetric. This follows directly, since we can revert the orientation of all paths P appearing in the above sums. This implies that local (edge-wise) coordinates change from ξ to ||- ξ and analogously for '. If we then use that c()^-1𝕋_P = c(')^-1𝕋_-P it follows immediately from (<ref>) that K_f is symmetric when all these operations are jointly executed. 

The kernel K_f on the graph is such that it coincides with the kernel on individual edges and takes scattering at the vertices into account when linking different edges together.

This allows us to define a convolution for f ∈ℒ^1, suitable (to be specified later) functions u on 𝒢, and x ∈𝒢 as

    (f*_𝒢 u)(x):=∫_𝒢 K_f(x,y)u(y)  dy

satisfying the associative property, see Lemma <ref>, for f,g∈ℒ^1

    (f*_𝒢(g*_𝒢u))(x) = ((f*g)*_𝒢u)(x).









[Convolution semigroup on 𝒢]
To extend the definition of a convolution semigroup on  with kernel k_t ∈ℒ^1, for t>0, to the graph 𝒢, we make the Ansatz

    (T_tf)(x) !=∫_𝒢 K_k_t(x,y) f(y)  dy,     x∈𝒢.

If the choice of kernel k_t is clear, we may just write K_t.





 §.§ Convergence properties of convolution semigroups

To analyze the semigroup (<ref>), we now introduce operators 


    (E^(ν)_f)(',ξ)   :=K_f^(ν)((,0),(',ξ)),  ∈ℰ(𝒢), (',ξ)∈𝒢, ν∈{1a,1b,1,2a,2b,2}.

 
 To link the function E_f(',ξ')=K_f((,0),(',ξ')) to the kernel K_f((,ξ),(',ξ')) in the subsequent Lemma, we define the shift operator (τ_ξ f)(x):=f(x-ξ). Then the shift ∋ s ↦τ_s f=f(∙-s) ∈ℒ^1 is continuous with ‖τ_s ‖_ℒ(ℒ^1)≤κ for all s ∈ [-ℓ_↑,ℓ_↑] and some κ>0. 

 
 [L^1-L^∞ estimate]
 
 For u ∈ L^∞(𝒢) and f ∈ℒ^1 we have
 
    sup_x ∈𝒢∫_𝒢| K_f(x,y) u(y)| dy  ≲κ‖ f ‖_ℒ^1‖ u ‖_L^∞.

In particular, for all ∈ℰ(𝒢) the operator E_ defines a continuous operator E_∈ℒ(ℒ^1, L^1(𝒢)) with bound ‖ E_ f ‖_L^1≲κ‖ f ‖_ℒ^1 independent of . 
 
 
The statement follows directly by applying (<ref>) to (<ref>), substituting the argument of f in (<ref>) as the new integration variable, such that

    ∫_𝒢| K_f(x,y) u(y)| dy    ≲κ/c()∑_m ∈ℕ_0 
    ' ∈ℰ(𝒢) 
     P ∈𝒫_,'(m) c(') |𝕋_P|‖ f ‖_L^1(I_m)‖ u ‖_L^∞
       ≲κ∑_m ∈ℕ_0 3^m ‖ f ‖_L^1(I_m)‖ u ‖_L^∞= κ‖ f ‖_ℒ^1‖ u ‖_L^∞.

This concludes the proof.
 
 



Let f ∈ℒ^1. The linear convolution maps, on spaces of continuous functions, f*_𝒢: L^∞(𝒢) → BUC(𝒢) and f*_𝒢:C_0(𝒢) → C_0(𝒢)
are bounded. In addition, the convolution f*_𝒢: L^p(𝒢) → L^p(𝒢) is a bounded map satisfying 

    ‖ f*_𝒢 u ‖_L^p(𝒢)≤ (κ‖ f ‖_ℒ^1) ‖ u ‖_L^p(𝒢) for all p∈ [1,∞].



The continuity of the shift is immediate if we assume that f ∈ C_c(). We therefore start by showing that the convolution is uniformly continuous: For η,η' ∈ [0,| e |] with |η-η' |≤min{δ, ℓ_↓}, where δ is chosen such that ‖ (τ_η-τ_η') f ‖_ℒ^1+‖ (τ_-η-τ_-η') f ‖_ℒ^1≤ε, and Lemma <ref>

    | (f*_𝒢u)(η,)-(f*_𝒢u)(η',) |
       = |∑_' ∈ℰ(𝒢) c(') ∫_0^|' |(E^(1)_(τ_η f)(ξ,')-E^(1)_(τ_η' f)(ξ,'))u(ξ,')  dξ|
          + |∑_' ∈ℰ(𝒢) c(') ∫_0^|' |(E^(2)_(τ_-η f)(ξ,')-E^(2)_(τ_-η' f)(ξ,'))u(ξ,')  dξ|
       ≤‖ E^(1)_(τ_η f)-E^(1)_(τ_η' f) ‖_1 ‖ u ‖_∞ + ‖ E^(2)_(τ_-η f)-E^(2)_(τ_-η' f) ‖_1‖ u ‖_∞
       ≤( ‖ (τ_η-τ_η') f ‖_ℒ^1+‖ (τ_-η-τ_-η') f ‖_ℒ^1)‖ u ‖_∞≤ε‖ u ‖_∞.

The result then follows since C_c() is dense in ℒ^1.
To study the map on C_0(𝒢), it suffices to use the density of C_c() in ℒ^1 and of C_c(𝒢) in C_0(𝒢). This immediately implies that f*_𝒢 maps C_0(𝒢) into C_0(𝒢) by Lemma <ref>.

It remains therefore to show L^p-boundedness:

To show boundedness on L^p(𝒢)-spaces that satisfy p<∞, we use that for p^-1+q^-1=1 by Hölder's inequality, Lemma <ref>, and the symmetry of the integral kernel 

    ∫_𝒢| K_f((',ξ'),(,ξ)) || u(,ξ) | dλ_c(,ξ) 
       =∫_𝒢| K_f((',ξ'),x) |^1/q (| K_f((',ξ'),x) || u(x) |^p)^1/p dx 
       ≤‖ K_f((',ξ'),∙) ‖_1^1/q(∫_𝒢| K_f((',ξ'),x) || u(x) |^p  dx)^1/p
       ≲ (κ‖ f ‖_ℒ^1)^1/q(∫_𝒢| K_f((,ξ),x) || u(x) |^p  dx )^1/p.

Hence, we get, using again Lemma <ref>

    ‖ f*_𝒢u ‖_p^p    ≲ (κ‖ f ‖_ℒ^1)^p/q∫_𝒢∫_𝒢| K_f(x,y) || u(x) |^p  dx  dy 
       ≲ (κ‖ f ‖_ℒ^1)^p/q∫_𝒢‖ E_(τ_-ξ f) ‖_1 | u(e,ξ) |^p dλ_c(e,ξ) 
       ≲ (κ‖ f ‖_ℒ^1)^p/qκ‖ f ‖_ℒ^1‖ u ‖_p^p≤ (κ‖ f ‖_ℒ^1)^p‖ u ‖_p^p.

This concludes the proof.




Let t↦ k_t ∈ℒ^1 be a differentiable function with ∂ k_t/∂ t∈ℒ^1, such that for all time t>0 we have ∫_ k_t=const and ∫_𝒢 K_∂ k_t/∂ t(x,y)  dy =0. Assume that for every ε>0 we have
lim_t ↓ 0∑_m ∈ℕ_0 3^m‖ k_t ‖_L^1(I_m \ [0,ε)) =0 and there exists a function g_ε∈ℒ^1 with | k_t(x) |≤ g_ε(x) for all x ∈\(-ε,ε) and sufficiently small times. Then, it follows that for every x ∈𝒢 on the interior of an edge 

    ∫_𝒢 K_k_t(x,y)  dy= const.



Let x=(, ξ) be a point in the interior of  then by assumption there is α_x=12min{d(x,i()),d(x,t())}>0 such that G_x:=𝒢\{x-α_x,x+α_x}

    lim_t ↓ 0∫_G_x K_k_t(x,y)  dy = 0  and lim_t ↓ 0∫_{x-α_x,x+α_x} K_k_t(x,y)  dy =lim_t ↓ 0∫_ℝ k_t(s)  ds=const.

We then observe that by the dominated convergence theorem 

    ∂/∂ t∫_𝒢 K_k_t(x,y)  dy = ∫_𝒢K_∂ k_t/∂ t(x,y)  dy =0.

Thus, it follows that ∫_𝒢 K_t(x,y)  dy= ∫_ℝ k_t(s)  ds= const.


We proceed with a result on approximate identities on metric graphs 𝒢 that will allow us to show strong continuity of C_0-semigroups.

Consider a kernel k_t as in Lemma <ref>. Then, we obtain the following strong limits for the semigroup defined by T_tu:= k_t*_𝒢 u

    lim_t ↓ 0‖ k_t*_𝒢 u - u ‖_∞=0     for all u ∈ C_0(𝒢), BUC(𝒢)  and 
       lim_t ↓ 0‖ k_t*_𝒢 u - u ‖_p=0,    for all u ∈ L^p(𝒢), p∈ [1,∞).



The dominated convergence theorem implies the existence of the following limit

    lim_t ↓ 0sup_ξ' ∈ [0,ℓ_↓]∑_m = 2^∞ 3^m∫_0^ℓ_↓| k_t(ξ'+mℓ_↓-ξ) | dξ=0.










To simplify the notation, we write the kernel K_k_t for x ∈ and y ∈' as

    K_k_t(x,y):=   c()^-1( k_t(x-y)δ_, '+∑_ m ∈ℕ
     P ∈𝒫_,'(m)𝕋_P k_t(| x-t(±) |+ | P | + | y-i(±')|) )

where the summation is also over all four possible orientation of the initial and terminal edges. 

We then start by showing the uniform convergence for u ∈ BUC(𝒢), which also implies the result for C_0(𝒢). Using the uniform limit (<ref>), we find that

    | (k_t*_𝒢 u)(ξ,)- u(ξ,) |≤∫_0^||| k_t(ξ-ξ') || u(ξ',)-u(ξ,) | dξ'+ ∑_m ∈ℕ
    ' ∈ℰ(𝒢) 
     P ∈𝒫_,'(m)c(')|𝕋_P |/c()×
       ∫_0^|' || k_t(| (ξ,)-t(±) |+ | P | -||+ | (ξ',')-i(±')|) || u(ξ',')-u(ξ,) | dξ' 
       ≤∫_0^||| k_t(ξ-ξ') || u(ξ',)-u(ξ,) | dξ' + 2‖ u ‖_∞ε + ∑_' ∈ℰ(𝒢) 
     P ∈𝒫_,'(1)c(')|𝕋_P |/c()×
       ∫_0^|' || k_t(| (ξ,)-t(±) |+ | P | -||+ | (ξ',')-i(±')|) || u(ξ',')-u(ξ,) | dξ'

for all t ∈ (0,t_0) sufficiently small and (ξ,') ∈𝒢.


We then split the integral in the second-to-last line above into an integral over a set I_δ:=(ξ-δ,ξ +δ) ∩ (0,||) and its relative complement I_δ^C:=(ξ-δ,ξ +δ)^C ∩ (0,||). 
We see that for ξ∈ I_δ^C, the integral in the second-to-last line, and the integral in the last line are small in virtue of the limit lim_t ↓ 0∫_\ (-δ,δ)| k_t(y)| dy =0.
If ξ∈ I_δ, then | u(ξ',)-u(ξ,) | is small by the uniform continuity of u. Hence, we have shown that 

    lim sup_t ↓ 0‖ k_t*_𝒢 u-u ‖_∞ =0.


To show general L^p convergence, we start by showing L^1 convergence, first. It suffices by Lemma <ref> to show by density the convergence for u ∈ C_c(𝒢) and thus we now fix R>0 large enough, such that u(x):=0 for d(x,o) ≥ R where o is a fixed reference point of 𝒢. Then we first consider x=(ξ,) with d(x,o) > 2R. It follows that the expression for the convolution reduces to

    (k_t*_𝒢 u)(x)=∑_m ≥ 2 
    ' ∈ℰ(B_R+ℓ_↑(o) ) 
     P ∈𝒫_,'(m)c(')/c()𝕋_P ∫_0^|' | k_t(ξ'+| P |-ξ) u(ξ',')  dξ'

where B_R+ℓ_↑(o) is the ball with respect to the graph metric.

This implies that for any small ε>0 we have for t small enough that | (k_t*_𝒢 u)(x)- u(x) | = |(k_t*_𝒢u)(x) |≤ε by (<ref>) for x satisfying d(x,o) ≥ 2R.

On the other hand, for x satisfying d(x,o) ≤ 2R the convergence is uniform by (<ref>) and thus also in L^1. 

For functions g ∈ L^1 ∩ L^∞ we have ‖ g ‖_p≤‖ g ‖^1/p_1‖ g ‖_∞^(p-1)/p.
Applying this to g=u-k_t*_𝒢 u, and using the convergence in both L^1-norm and L^∞-norm, we conclude convergence in L^p-norm for any intermediate exponent p ∈ (1,∞), whence the claim. Thus, let p < ∞, u ∈ L^p and ε>0 small. Choose v ∈ C_c such that ‖ u-v‖_p< ε. Then by (<ref>) and by Lemma <ref>

    lim sup_t ↓ 0‖ k_t*_𝒢 u-u ‖_p 
        ≤lim sup_t ↓ 0[ ‖ k_t*_𝒢 v-v‖_p + ‖ k_t*_𝒢(u-v) ‖_p ]+ ‖ u-v ‖_p 
       ≤ (C+1) ‖ u-v ‖_p ≤ (C+1)ε.

This concludes the proof.



 §.§ Boundary conditions for convolutions on graphs


We now show that the convolution, introduced in (<ref>), is compatible with the boundary conditions introduced in Section <ref>, i.e., the convolution is continuous at the vertices and its derivative satisfies the natural condition (<ref>) at the vertices. 


Let f ∈ℒ^1 then it follows that for any u ∈BUC(𝒢)

    f*_𝒢 u  is continuous at the vertices .


Let f ∈ℒ^1 be differentiable such that | f' |∈ℒ^1 then it follows that for any u ∈BUC(𝒢)

    f'*_𝒢 u  satisfies a Kirchhoff condition at the vertices .



To start with the continuity conditions, we fix an edge , then the kernel K^(1), defined in (<ref>), yields using Def. <ref>

    K^(1)_f((,ξ),(',0)):=c( )^-1( δ_, ' f(ξ)+∑_m ∈ℕ( ∑_P ∈𝒫_,'(m)𝕋_P f(| P |-ξ) 
                + ∑_P ∈𝒫_,-'(m)𝕋_P f(|' | +| P |-ξ) ) ) 
       =c( )^-1( δ_, ' f(ξ)+𝕋_,' f(||-ξ) +∑_m ∈ℕ( ∑_”∈ t^-1(i(')) 
     P ∈𝒫_,”(m)𝕋_P (2c(”)/c(i(')) - δ_”,-')
            f(|”|+ | P |-ξ)+ ∑_P ∈𝒫_,-'(m)𝕋_P f(|' | +| P |-ξ) ) ) 
       =c( )^-1( δ_, ' f(ξ)+ 𝕋_,' f(||-ξ) + ∑_m ∈ℕ
    ”∈ t^-1(i(')) 
     P ∈𝒫_,”(m)𝕋_P 2c(”)/c(i(')) f(|”|+ | P |-ξ) ).

By an analogous computation, we find for the remaining part of the kernel in (<ref>)

    K^(2)_f((,ξ),(',0)):=c( )^-1( 𝕋_-,'f(ξ)+ ∑_m ∈ℕ
     P ∈𝒫_-,-'(m) 𝕋_P f(|' |+| P |-(||-ξ)) 
          +∑_m ∈ℕ
    ”∈ t^-1(i(')) 
     P ∈𝒫_-,”(m)𝕋_P (2c(”)/c(i(')) - δ_”,-') f( |”| + | P |-(||-ξ))) 
         =c( )^-1( ∑_”∈ t^-1(i(')) 
     P ∈𝒫_-,”(m)𝕋_P 2c(”)/c(i(')) f(|”| + | P |-(||-ξ)) + 𝕋_-, 'f(ξ) ).

To see that this representation implies continuity at vertices, we introduce coefficients, that only depend on the terminal vertex ∈𝒱 but not(!) on the adjacent edge

    (α_0(,i())f )(ξ) = 2c()/c(i()) f(ξ)  and (α_0(,t())f)(ξ) = 2c()/c(t()) f(||-ξ)
 
and otherwise for m ∈ℕ

    (α_m(,)f)(ξ)   :=2 ∑_”∈ t^-1{v }
     P ∈𝒫_,”(m)𝕋_P c(”)/c() f(| P |+ |”|-ξ) 
          + 2 ∑_”∈ t^-1{v }
     P ∈𝒫_-,”(m)𝕋_P c(”)/c() f(|”|+| P |- (||-ξ)) .

 From (<ref>) and (<ref>) and an analogous calculation for terminal vertices, we find that 
 
    K_f((,ξ),) = c()^-1∑_m ∈ℕ_0 (α_m(,)f)(ξ).

 
To derive the natural condition at the vertices, we find for derivatives of K^(1)_f and K^(2)_f in (<ref>) and some fixed edge , using that since f is an even function, its derivative is an odd function

    d/dξ'|_ξ'=0 K^(1)_f((,ξ),(',ξ')):=c( )^-1(- δ_, ' f'(ξ)
          +∑_m ∈ℕ( ∑_P ∈𝒫_,'(m)𝕋_P f'(| P |-ξ)- ∑_P ∈𝒫_,-'(m)𝕋_P f'(|' | +| P |-ξ) ) ) 
       =c( )^-1( -δ_, ' f'(ξ)+ 𝕋_,' f'(||-ξ) 
          + 2 ∑_m ∈ℕ
    ”∈ t^-1(i(')) 
     P ∈𝒫_,”(m)𝕋_P (c(”)/c(i(')) - δ_”,-')f'(|”|+ | P |-ξ) ).

Similarly, we find that

    d/dξ'|_ξ'=0 K^(2)_f((,ξ),(',ξ')):=c( )^-1( ∑_m ∈ℕ
    ”∈ t^-1(i(')) 
     P ∈𝒫_-,”(m)𝕋_P (2c(”)/c(i(')) - δ_”,-') ×
       f'( |”| + | P |-(||-ξ)) + 𝕋_-,'f'(ξ) +∑_m ∈ℕ
     P ∈𝒫_-,-'(m) 𝕋_P f'(|' |+| P |-(||-ξ))) 
         =c( )^-1( 2 ∑_m ∈ℕ
    ”∈ t^-1(i(')) 
     P ∈𝒫_-,”(m)𝕋_P (c(”)/c(i('))- δ_”,-') f'( |”| +| P |-(||-ξ)) + 𝕋_-, 'f'(ξ) ).

The study of directional derivatives at terminal ends is fully analogous.



From the identity

    ∑_∈ℰ_ c() ( c(')/c() -δ_',-) = c(')-c(')=0,

we infer that 

    ∑_' ∈ℰ_i(') c(') ∑_”∈ t^-1(i(')) 
     P ∈𝒫_-,”(m) 2𝕋_P (c(”)/c(i('))- δ_”,-')=0.


We then define for edges ' ∈ℰ(𝒢) with i()=i(') the quantity (β_0(,')f)(ξ) := 2 ( c()/c(t()) -δ_,')f'(ξ) and for edges ' ∈ℰ(𝒢) with t()=i(') the expressions 

    (β_0(,')f)(ξ) := 2 c()/c(t()) f'(||-ξ).

Moreover, for m ≥ 1

    (β_m(,')f)(ξ)    := 2 ∑_m ∈ℕ
    ”∈ t^-1(i(')) 
     P ∈𝒫_,”(m)𝕋_P (c(”)/c(i(')) - δ_”,-')f'(|”|+ | P |-ξ) 
       +2 ∑_m ∈ℕ
    ”∈ t^-1(i(')) 
     P ∈𝒫_-,”(m)𝕋_P (c(”)/c(i('))- δ_”,-') f'( |”| +| P |-(||-ξ)).

It then follows that the convolution satisfies the Kirchhoff condition (<ref>) due to

    ∑_' ∈ℰ_ c(') ∂ K_f((,ξ),(',0))∂
    n_ = c()^-1∑_' ∈ℰ_ c(') (β_m(,')f)(ξ)= 0.

This concludes the proof.






 §.§ Convolution semigroups


Let us continue by studying the heat equation (or more generally a convection-diffusion problem) as well as higher-order parabolic equations using our integral kernel framework on infinite metric graphs. 
For equations

    ∂_t u(t) = Δ u(t,x) + ξ u(t,x)  and ∂_t v(t,x) =-(- Δ)^m v(t,x)
 
the solutions on  are given by convolutions u(t) = e^ξ t h_t*u_0 and v(t) = k_t*v_0.
Here, h_t is the standard heat kernel on  given by h_t(x) =  e^-x^2/4t/(4π t)^1/2 
and the kernel k_t, see <cit.>, satisfies estimates

    | k_t(x,y) |≤ c_1 t^-1/(2m)exp(-c_2 | x-y |^2m/2m-1/t^1/(2m-1)+c_3t)

for some c_1,c_2,c_3>0, cf. <cit.> and <cit.>.



Let 𝒢 be a metric graph and consider the convolution defined in (<ref>) and the integral kernels (h_t)_t>0 and (k_t)_t>0 defined in Section <ref>. Let also ξ∈.
Then the following assertions hold.



  * The C_0-semigroup (T_tf)(x)=e^ξ t (h_t*_𝒢f)(x) 

    ∂_t v(t,x) = Δ v(t,x) + ξ v(t,x),    v(0,x)=f(x)

on each of the following spaces:


  * L^p(𝒢), p ∈ [1,∞); 

  * C_0(𝒢);

  * BUC(𝒢).
 



  * The C_0-semigroup (T_tf)(x)=e^ξ t(k_t*_𝒢f)(x) satisfies the equation 

    ∂_t v(t,x) =-(- Δ)^m v(t,x)+ ξ v(t,x),    v(0,x)=f(x)

on each of the following spaces:


  * L^p(𝒢), p ∈ [1,∞); 

  * C_0(𝒢);

  * BUC(𝒢).
 
	
 
Additionally, at the vertices  of 𝒢 the function v(t):=T_t f satisfies in either case


  * continuity conditions on (Δ^mv)(t,), t>0 and m ∈ℕ_0;

  * Kirchhoff (i.e., zero sum) conditions on (∂/∂ nΔ^mv)(t,), t>0 and m ∈ℕ_0.




For the heat kernel it is obvious that Δ h_t ∈ℒ^1. The kernel k_t is explicitly given, for some normalization constant α_m by the formula

    k_t(x):=α_m/t^1/2m∫_0^∞ e^-s^2mcos(xs/t^1/2m)  ds.

Differentiating this kernel in time shows that 

    Δ^m k_t(x):=α_m/t^1/2m∫_0^∞e^-s^2m s^2mcos(xs/t^1/2m)/t ds.

It then follows since k_t is the Fourier transform of an entire function that is integrable on any slice ℝ+iη for η∈, that k_t decays faster than any exponential, i.e., for any b>0 it follows that |Δ^m k_t(x) |≤ C_b e^-b | x |.
This implies that also Δ^m k_t ∈ℒ^1. In either case, this immediately implies that v(t,x):=T_tf(x) solves the respective equation on the edges. That the solution satisfies the correct boundary conditions follows from the symmetry of the kernels h_t,k_t (even functions) and Lemma <ref>. More precisely, Δ^m h_t, Δ^m k_t are even functions in ℒ^1 for all m ∈ℕ_0 and thus the Lemma applies to every order.


We will postpone the identification of the generator with (<ref>) until Section <ref> and, in particular, Proposition <ref> (heat semigroups) and Theorem <ref> (higher order parabolic problems).




§ SPECTRAL INDEPENDENCE AND SCHRÖDINGER SEMIGROUPS



We start by analyzing mapping properties of Schrödinger semigroups. The ultracontractivity estimates of the following subsection will also imply a crucial role in the proof of spectral independence in the subsequent subsection.


 §.§ Ultracontractivity and Schrödinger semigroups



We consider Schrödinger operators H=-Δ + V with potentials V ∈ L^∞(𝒢). We also introduce the operator H_-=-Δ + V_- where V_- is the negative part of V.
Consider the Hilbert space L^2(𝒢) and the following closed symmetric form 

    τ(u,v) := ⟨ H u ,v ⟩_L^2 + c ⟨ u,v ⟩_L^2

where c is such that τ is a positive form. 
We then define another form 

    τ_-(u,v) := ⟨ H_- u ,v ⟩_L^2 + c ⟨ u,v ⟩_L^2.

Using that ∇| u | = (u) ∇ u, we find for τ [u]:=τ(u,u) that τ satisfies the first Beurling-Deny criterion τ[ | u |] ≤τ [u] ,  for  u ∈ H^1(𝒢).
Moreover, we have the domination of forms τ_-(u,v) ≤τ(u,v), for 0≤ u,v ∈ H^1(𝒢).
Thus, we have | e^-tHf |≤ e^-tH_-| f | for all t≥ 0 and all f ∈ L^2(𝒢).



This implies that to study L^q-L^p smoothing of the Schrödinger semigroup (e^-tH_p)_t ≥ 0 it suffices to study T_p(t):=e^-tH_p- where the index p indicates the reference space L^p(𝒢). To simplify the notation we will just write H instead of H_- and V instead of V_- in the sequel. We will also write T_p^κ(t):=e^-t (-Δ_p+κ V_-) for the semigroup generated by Δ_p-κ V_- for some κ∈.


[Ultracontractivity] 

Let 𝒢 be a metric graph. Then the following assertions hold.



  * Let V∈ L^∞(𝒢). Then the semigroup (T(t))_t≥ 0=(e^t(Δ+V))_t≥ 0 consists of operators that are bounded from L^q(𝒢) to L^p(𝒢) for all p ∈ [q,∞] and q ∈ [1,∞); we denote their realization in L^q(𝒢) by T_q(t).

If we additionally impose Assumption <ref>, then there exists M=M(q,p)>0 such that

    ||T_q(t)f||_L^p≤ Mt^-1/2(q^-1-p^-1)e^tV^+_∞||f||_L^q  for all t>0.



  * Let m≥ 2. Then the semigroup (S(t))_t≥ 0=(e^-t(-Δ)^m)_t≥ 0 consists of operators that are bounded from L^q(𝒢) to L^p(𝒢) for all p ∈ [q,∞] and q ∈ [1,∞); we denote their realization in L^q(𝒢) by S_q(t).


If we additionally impose Assumption <ref>, then there exists M=M(q,p)>0 such that

    ||S_q(t)f||_L^p≤ Mt^-1/2m(q^-1-p^-1)||f||_L^q  for all t>0.





We introduce the constant C_t:=‖ T^0_1(t) ‖_ℒ(L^1,L^∞) for t>0. This quantity is finite as the integral kernel (<ref>) is uniformly bounded, which follows from (<ref>) and the finiteness of the ℒ^1-norm. Let κ>1 such that κ^-1+η^-1=1 for some η∈ℕ.

Since the negative part of the potential is relatively L^1 zero-bounded, it follows that the Schrödinger operator H^κ_1:=-Δ_1+κ V generates a semigroup T^κ_1(t) on L^1(𝒢) such that for some M>0 and ω∈ℝ we have ‖ T_1^κ‖_ℒ(L^1)≤ M e^ω t.

Let F_t(r):=T^r κ_2(t) ∈ℒ(L^2), then by <cit.> and positivity of T_2^r κ(t/n) we have for r ∈ [0,1]

    |(e^ i s κ Vt/n T_2^r κ(t/n) )^n f |≤ T_2^r κ(t) | f |.


By the Trotter–Kato formula we have from taking the limit n →∞ in the above equation | T_2^(r+is)κ(t) f |≤ T_2^rκ(t) | f |.
This implies that 

    ‖ F_t(is) f ‖_L^∞   ≤‖ F_t(0) | f |‖_L^∞≤‖ F_t(0) ‖_ℒ(L^1,L^∞)‖ f ‖_L^1 and 
    ‖ F_t(1+is) f ‖_L^1   ≤‖ F_t(1) | f |‖_L^1≤‖ F_t(1) ‖_ℒ(L^1,L^1)‖ f ‖_L^1

such that 

    ‖ F_t(is) ‖_ℒ(L^1,L^∞)   ≤‖ F_t(0) ‖_ℒ(L^1,L^∞)=C_t  and 
    ‖ F_t(1+is) ‖_ℒ(L^1,L^1)   ≤‖ F_t(1) ‖_ℒ(L^1,L^1)≤ M e^ω t.


Stein's interpolation theorem then implies that ‖ F_t(s) ‖_ℒ(L^1,L^s^-1)≤ C_t^1-s(Me^ω t)^s
such that by choosing s= κ^-1 we find ‖ F_t(s) ‖_ℒ(L^1,L^κ)≤ C_t^η^-1(Me^ω t)^κ^-1.
The Riesz–Thorin theorem implies then by interpolating T_1(t): L^1(𝒢) → L^κ(𝒢) and T_1(t)^*: L^η(𝒢) → L^∞(𝒢) with η∈ℕ that T_1(t) induces bounded operators T_q(t): L^q(𝒢) → L^p(𝒢) with q^-1-p^-1=η^-1.

We then have that for q_0=1 < q_1<...<q_η=∞ with q_j-1^-1-q_j^-1=η^-1

    ‖ T_1(t) ‖_ℒ(L^1,L^∞)≤∏_j=1^η‖ T_1(t/η) ‖_ℒ(L^q_j-1,L^q_j)≤ C_t/η M^η-1e^tω(η-1).


For higher-order parabolic systems the L^q → L^∞ boundedness of (S(t)) follows from the ℒ^1-boundedness of the integral kernel. Thus, the Riesz–Thorin theorem implies the L^q → L^p boundedness of (S_q(t)).


In order to obtain estimates (<ref>) and (<ref>) we observe that by Theorem <ref> we know that the semigroup (T_2(t)) is analytic on L^2(𝒢). Its generator, that we denote here by H_1=Δ-V, is associated with the quadratic form

    a_1(u)   =∫_𝒢 |u'(x)|^2 dx+∫_𝒢V(x)|u(x)|^2 dx,     u∈ H^1(𝒢).

 Moreover, (S_2(t)) on L^2(𝒢) is analytic and is generated by H_m:=-(-Δ)^m for m≥2 with domain

    D(H_m)={u∈ H^2m(𝒢) : u^(2n)∈ C(𝒢),∑_∈ℰ_(𝒢) ∂∂
    n_u^(2n)_() =0, 0≤ n<m}.

It is easy to see (adapt the proof of <cit.>) that this operator is associated with the quadratic form 

    a_m(u)   =∫_𝒢|u^(m)(x)|^2  dx,
    
    D(a_m)   ={u∈ H^m(𝒢) : u^(2n)∈ C(𝒢), ∑_∈ℰ_(𝒢) ∂∂
    n_u^(2n)_() =0, 0≤ n<⌊m/2⌋}.

Let u∈ H^1(𝒢), then an application of Gagliardo-Nirenberg inequality (<ref>) gives

    ||u||_L^∞   ≤ c_1||u'||_L^2^1/2||u||_L^2^1/2
       ≤ c_1(a_1(u)+||V||_L^∞||u||_L^2^2)^1/4||u||_L^2^1/2
       ≤c̃_1 a_1(u)^1/4||u||_L^2^1/2+c̃_2||u||_L^2.

Let now u∈ H^m(𝒢), m≥2, again an application of (<ref>) gives

    ||u||_L^∞   ≤ c_1||u^(m)||_L^2^1/2m||u||_L^2^2m-1/2m
       =c_1a_m(u)^1/4m||u||_L^2^2m-1/2m.

Now we let u=P(t)f, where P(t)=T_2(t) for m=1 and P(t)=S(t) for m≥2. By (<ref>), (<ref>) and using contractivity (quasi contractivity for (T_2(t))) and analyticity on L^2 and Cauchy–Schwarz inequality, it follows that for all t > 0

    ||P(t)f||_L^∞   ≤ k_1⟨ -H_mP(t)f,P(t)f⟩^1/4m||P(t)f||_L^2^2m-1/2m+k_2||P(t)f||_L^2
       ≤ k_1e^ct||V_+||_L^∞||H_mP(t)f||_L^2^1/4m||P(t)f||_L^2^1/4m||f||_L^2^2m-1/2m+k_2e^ct||V_+||_L^∞||f||_L^2
       ≤ e^c̃t(k_1t^-1/4m+k_2)||f||_L^2
       ≤ Mt^-1/4me^ω t||f||_L^2.

Observe that in the case m≥ 2 the constants c,c̃,k_2,ω vanish.
Now, applying this last inequality to the semigroup generated by H_m^*, by duality this yields the same bound to hold for the L^1-L^2 norm as well and then the claimed estimates follow for q=1,p=∞. Riesz–Thorin interpolation between (1,∞) and (2,2) implies that the result holds also for q,q' where q^-1+q'^-1=1. Another application of Riesz–Thorin interpolation between (q,q) and (q,q') yields then the result for pairs (q,p).




 §.§ Spectral independence of Schrödinger operators and -(-Δ)^m




From the ultracontractivity proved in Theorem <ref>, let us now prove the p-independence of the spectrum of generators.
[L^p-independence]

Let 𝒢 be a metric graph that satisfies Assumption <ref>. The spectrum of the (negative) Schrödinger operator H_p:=Δ_p-V, with V ∈ L^∞(𝒢), on L^p(𝒢) with p ∈ [1,∞) and Laplacians of higher order H_p =- (-Δ_p)^m on L^p(𝒢) with m ≥ 1, does not depend on p, i.e., σ(H_2)=σ(H_p) ⊂. The inclusion σ(H_2)⊂σ(H_p) holds also without any growth assumption on the graph. Moreover, the resolvents of operators H_p are consistent. 




“σ(H_2) ⊂σ(H_p)”
Let p ∈ [q,∞], then T_q(t) ∈ℒ(L^q(𝒢),L^p(𝒢)∩ L^q(𝒢)) for t >0, we have for x ∈ D(H_q)

    H_pT_q(t)x    =lim_h ↓ 0T_p(h)-id/h T_q(t)x = lim_h ↓ 0T_q(h)-id/h T_q(t)x 
       = lim_h ↓ 0 T_q(t)T_q(h)-id/hx = T_q(t)H_qx.

Let λ∈ρ(H_q) ∩ρ(H_p), then from (<ref>) we conclude that

    (λ-H_p)^-1T_q(t) 
       = (λ-H_p)^-1 T_q(t)(λ-H_q)(λ-H_q)^-1
       = (λ-H_p)^-1 (λ-H_p)T_q(t)(λ-H_q)^-1 = T_q(t)(λ-H_q)^-1.

Taking the limit t ↓ 0, we obtain that resolvents are consistent

    (λ-H_p)^-1|_L^q ∩ L^p =(λ-H_q)^-1|_L^q ∩ L^p.

By duality, the same is true for 2 ≤ p ≤ q ≤∞.
Thus, for q^-1+q'^-1=1 and λ∈ρ(H_q)=ρ(H_q')

    (λ-H_q')^-1|_L^q ∩ L^q' = (λ-H_q)^-1|_L^q ∩ L^q'.


For x ∈ L^p(𝒢) ∩ L^q(𝒢), (<ref>) implies 

    lim_t ↓ 0 T_q(t)(λ-H_q)^-1 x = (λ-H_q)^-1 x  and 
       lim_t ↓ 0 (λ-H_p)T_q(t)(λ-H_q)^-1 x = lim_t ↓ 0 T_q(t) x=x.

Using that λ-H_p is closed implies then that (λ-H_p)(λ-H_q)^-1 x = x
and thus, we conclude (λ-H_q)^-1 x = (λ-H_p)^-1x.
It remains to notice then that by the Riesz–Thorin theorem, by interpolating between (λ-H_q')^-1 and (λ-H_q)^-1, it follows that (λ-H_q)^-1 is also continuous on L^p which implies that (λ-H_p) is surjective. It is also injective, since otherwise there is a non-zero u ∈ D(H_p) such that H_p u = λ u and by (<ref>)

    H_q' T_p(t)u = T_p(t)H_p u = λ T_p(t) u

which contradicts λ∈ρ(H_q').



“σ(H_p) ⊂σ(H_2)”: 
To show the converse implication, we use that the resolvents are consistent for z ∈ρ(H_2) ∩ρ(H_p) by (<ref>).
We then notice that the resolvent z↦ (H_2-z)^-1|_L^p ∩ L^2 extends on ρ(H_2) to a bounded operator on L^p which follows from the Combes–Thomas estimate in Proposition <ref>. By unique continuation, this extension is the resolvent (H_p-z)^-1.











We now continue by extending our results to spaces C_0(𝒢) and BUC(𝒢). 
We start with two far-reaching results from semigroup theory on the spectrum of the (higher-order) Laplacians -(-Δ)^m for m ≥ 1 and thereby provide an explanation why C_0(𝒢) and BUC(𝒢) fit naturally in the framework of L^p(𝒢)-spaces. 


Since duals of C_0-semigroups may not be strongly continuous, the concept of sun duals has been introduced to restore strong continuity. Given a C_0-semigroup (T(t)) on a Banach space X_c, its sun dual space is defined by 

    X_c^⊙:={x' ∈X_c'; lim_t ↓ 0‖ T(t)' x' -x' ‖_X_c'=0 }.

The sun dual semigroup is then defined as T(t)^⊙:=T(t)'|_X_c^⊙.
When it comes to spectral properties on different spaces, the relevance of this concept is due to the following result. 
<cit.>

Let (T(t)) be a strongly continuous semigroup with generator A and Y ⊂X_c^⊙ be a closed subspace that is invariant under T(t)'. If Y is equi-norming for X_c, i.e. for all x ∈X_c 

    ‖ x ‖_X_c= sup_y ∈ Y; ‖ y‖=1⟨ y,x ⟩,

 then ρ_∞(A)=ρ_∞(A'|_Y), where ρ_∞ is the connected component of the resolvent set containing a right half plane; and we have 
 
    (λ-A|_Y)^-1 = (λ-A)|_Y^-1     for all λ∈ρ_∞(A).


Moreover, the generator of T(t)'|_Y is A'|_ Y. 


For any g ∈ (L^1)^*=L^∞ the convolution with a kernel in ℒ^1 becomes by Lemma <ref> immediately a BUC(𝒢) function. Thus, in particular for semigroups defined in Theorem <ref>, L^1(𝒢)^⊙⊃ BUC(𝒢).
Theorem <ref> however, also applies to the closed subspace C_0(𝒢) of bounded uniformly continuous functions, since there is always a normalized sequence of C_0(𝒢) functions in this space converging a.e.  to (f) for any f ∈ L^1(𝒢).

Thus, we have established at this point already the following Corollary:


The connected component of the resolvent set containing a right half plane, that we denote by ρ_∞, satisfies for all m ∈ℕ_0 and V ∈ L^∞
 
    ρ_∞(-(-Δ_L^1)^m)=ρ_∞(-(-Δ_C_0)^m)=ρ_∞(-(-Δ_BUC)^m)  and 
       ρ_∞(Δ_L^1-V)=ρ_∞(Δ_C_0-V)=ρ_∞(Δ_BUC-V).

 In particular, let Assumption <ref> be satisfied. The spectrum of the Schrödinger operator H_p:=-Δ_p+V, with V ∈ L^∞(𝒢), on L^p(𝒢) with p ∈ [1,∞) and Laplacians of higher order H_p =- (-Δ_p)^m on L^p(𝒢) with m ≥ 1, does not depend on p, i.e., σ(H_2)=σ(H_p) ⊂ and coincides with the spectrum on C_0(𝒢) and BUC(𝒢).


The first part is immediate from Proposition <ref>. The second one follows by combining the first part and Theorem <ref>.

 


 §.§ Counterexamples to spectral independence


We now provide an example that shows that the spectrum on BUC(𝒢) can also differ from the spectrum on L^2(𝒢).

[Rapidly growing graph]


Consider the Laplacian Δ on the infinite graph 𝒢 illustrated in Figure <ref>.
Take an edge e=(n,d) of the graph, where we parametrize all edges connected to the same node n ∈, on the upper horizontal line, with depth d ∈_0 (number of edges away from the horizontal line) by

    ψ_(t) = s_λ(t)e^iθ(n+d) + e^iθ(n+d+1)(s_λ(1)s_λ(t)-c_λ(1)s_λ(t))/s_λ(1).

This defines a family of eigenfunctions Ψ_θ= (ψ_) in BUC(𝒢) parametrized by a parameter θ∈ [0,2π) such that 

    ΔΨ_θ =( arccos(2e^iθ+e^-iθ/3))^2 Ψ_θ.

However, (arccos(2e^iθ+e^-iθ/3))^2 takes also values with non-zero imaginary part. This is impossible for the self-adjoint Laplacian we defined on L^2(𝒢). 


While in the previous example the rapid growth rate of the graph 𝒢 caused the spectral degeneracy between spaces L^2(𝒢) and BUC(𝒢), it is the unboundedness of the potential in the following example that causes the spectrum to depend on the underlying space:

[Harmonic oscillator]
Let us start by defining the operator H = -Δ + V for positive V ∈ L^∞_loc, defined weakly as the operator associated with the form Q_H given by	

    Q_H(f)= ∫_𝒢| f'(x) |^2+V(x) | f(x) |^2  dx,      f∈ D(Q_H),
 where D(Q_H)={f∈ L^2(𝒢): Q_H(f)∈ L^2(𝒢)}. We then have that H has compact resolvent if V satisfies a suitable growth condition.


If V(x) ≥ c ⟨ x ⟩^κ, where ⟨ x ⟩:=(1+ | x |^2)^1/2 for some κ>0, c>0, and x such that d(x,0)>r for some fixed r>0, then the operator H = -Δ + V– defined weakly as the operator associated with the form Q_H – on L^2(𝒢) has compact resolvent.


To show that H has compact resolvent, it suffices to show that the form domain D(Q_H) is compactly embedded in L^2(𝒢). To see this, we first notice that

    lim_r →∞∫_d(x,0)>r| f(x)|^2  dx ≤lim_r →∞1/⟨ r ⟩^κ∫_d(x,0)>r⟨ x ⟩^κ| f(x) |^2  dx=0.

Then, since the first derivative is the generator of translations (τ_a f)=f(∙-a), this implies that lim_a → 0‖τ_a f - f ‖_L^2(𝒢)=0. The Fréchet–Kolmogorov theorem implies therefore that the embedding D(Q_H) ↪ L^2(𝒢) is compact.


 

The following Corollary, which follows from interpolation theory <cit.>, shows that spectral equivalence holds, under the conditions of Proposition <ref>, at least for p ∈ (1,∞):

If T_2(t)=e^-H_2t is compact for 0<t<∞, then e^-H_p t is compact for t ∈ (0,∞) and p ∈ (1,∞) and analytic. Moreover, σ(H_p)= σ(H_2).


We will now see that the spectral invariance can fail dramatically for p=1 as soon as the potential is unbounded:
Consider equilateral graphs 𝒢 that are embedded into ℝ^d. We then start by introducing the Schrödinger operator

    (H_oscu)(x)=(-Δ u)(x) + ω^2 ⟨ x,-i() ⟩^2 u(x)

where ⟨∙, ∙⟩ denotes the usual inner product.


Let us recall that the harmonic oscillator on ^d given as H_ℝ^n = -Δ_^n + ω^2 | x |^2
has a complete orthonormal basis induced by product states ψ_j(x) := ∏_i=1^d ψ_j_i(x_i) ∈ L^2(^d).
It is then easy to check that the above eigenfunctions are also, for indices j_k=j_i, eigenfunctions to (<ref>).
More precisely, let  be an edge in direction _i: then we have (H_oscψ)_(x_i)=λ_j_iψ_(x_ i ).

In contrast to this, it is known <cit.> that on L^1() the operator

    H_ℝ = -Δ_ + ω^2 | x |^2
 
has spectrum σ(H_ℝ) = { z ∈: (z)>0 } and the spectrum consists of eigenfunctions of multiplicity 2.

Taking any of these eigenfunctions φ_z ∈ L^1() we can define an eigenfunction φ_z(x) := ∏_i=1^d φ_z(x_i) to H_osc. This shows that that the L^1 spectrum of H_osc contains the right half plane. In particular, the semigroup generated by H_osc on L^1(𝒢) is not analytic.





§ PROPERTIES OF SCHRÖDINGER SEMIGROUPS






We can now give a sufficient condition for Schrödinger operators H=-Δ+V to be accretive. Thus, by the Lumer–Phillips theorem, -H generates a contractive semigroups on L^p-spaces as well as spaces C_0 and BUC. 

The Schrödinger operator H_X_c, with positive potential V that is relatively Δ_X_c bounded with relative Δ_X_c-bound <1, is accretive on X_c=L^p(𝒢) for all p ∈ [1,∞) as well as spaces X_c=C_0(𝒢) and BUC(𝒢). 
In particular, the heat semigroup constructed in Subsection <ref> is contractive on all spaces X_c= L^p(𝒢) with p ∈ [1,∞) and C_0(𝒢), BUC(𝒢) and generated by Δ with domain given in (<ref>).


The statement is clear on L^2(𝒢) by the spectral theorem.

Now, consider H_L^1. To show that this operator is accretive, we take f ∈ D(H_L^1) and define the functional

    φ_f(g):=∫_𝒢g(x)  φ(f)(x)  dx  with φ(f)(x)=(f(x)) ‖ f ‖_L^1

from the duality set, see <cit.>. 
It suffices then to show that φ_f(-H_L^1f) ≤ 0 in order to conclude by <cit.> that the generator H_L^1 is accretive, i.e. for all λ >0 

    ‖ (-H_L^1-λ)^-1‖_ℒ(L^1(𝒢))≤λ^-1,

cf. <cit.>.
To see this, it suffices to look at one fixed edge ∈ℰ(𝒢).
There, we assume that f_ has x_1,...,x_n (possibly none) zeros such that =(i(),x_1)_=:I_1∪...∪(x_n,t())_=:I_n+1.
Then, if (f|_I_i)=1 we have f'(x_i)≤ 0 and if (f|_I_i)=-1 we obtain f'(x_i)≥ 0. Thus, we have for i=2,..,n that

    φ_f(-H_L^1f)= ∫_I_i (Δ f)_(x) φ(f)_(x) - V_(x) | f_(x) |‖ f ‖_L^1 dx ≤ 0.

Hence, it follows that

    ∫_ (-H_L^1f)_(x) φ(f)_(x)dx    ≤∫_ (Δ f)_(x) φ(f)_(x)dx = ( ∂ f_/∂ n_(t(e))+∂ f_/∂ n_(i(e)) ).

The Kirchhoff condition implies then by summing over ∈ℰ(𝒢) that

    ∫_𝒢 - (H_L^1f)_(x) φ(f)_(x)dx ≤ 0.




Clearly, the Laplacian with boundary conditions as defined in (<ref>) does not have any eigenfunctions with positive eigenvalue on L^1(𝒢) or L^2(𝒢) (dissipativity/integration by parts). The heat semigroup (T_p(t)) introduced in Subsection <ref> maps by Lemma <ref> into D(-Δ_p), defined in (<ref>), and thus its generator A_p is necessarily a restriction of Δ_p. 

Thus for λ >0 large we have that the invertible operator A_1-λ is a restriction of the injective operator Δ_1-λ which implies that they both coincide A_1=Δ_1. 
Thus, the heat semigroup is contractive on L^1(𝒢) by the Lumer-Phillips theorem. This implies that Δ_1-λ is invertible for any λ>0. 

Hence, by duality, there does not exist an eigenfunction u ∈ D(Δ_BUC) satisfying (Δ_BUC-λ)u=0, as otherwise Δ_1-λ would not be surjective on L^1(𝒢).
We conclude from repeating the above calculation on spaces L^p using the functional φ_f(g) = ∫_𝒢 g(x) sgn(f(x)) | f(x) |^p-1‖ f ‖^2-p_p  dx that Δ_p is dissipative on all spaces L^p(𝒢) with p >1. 
Riesz–Thorin interpolation of the semigroup on L^1(𝒢) and L^2(𝒢) and duality implies then the contractivity of the semigroup on all spaces L^p(𝒢) with p ∈ [1,∞) and spaces C_0(𝒢), BUC(𝒢). 
We conclude, as above, that the generator of all these semigroups must be a restriction of the Laplacian we defined on these spaces in Section <ref>. 
The result for Schrödinger semigroups then follows from <cit.>. 

We remark that higher order parabolic semigroups studied in Theorem <ref> do not necessarily have to be contractive which is obvious already for 𝒢=:

[Non-contractivity of bi-Laplacian]
Consider the semigroup generated by the bi-Laplacian -Δ^2 on L^1(ℝ). In this case, it is well-known that the kernel k_t associated with the semigroup (e^-Δ^2 t) takes negative values on a set of positive measure whereas ∫_ℝ k_t =1. This implies that ‖ k_t‖_L^1>1 which implies sup_‖ u_0 ‖_L^1=1‖ e^-Δ^2 t u_0 ‖_1 = sup_‖ u_0 ‖_L^1=1‖ k_t*u_0 ‖_1= ‖ k_t ‖_1>1 that the associated semigroup on L^1 cannot be contractive.


In fact, aside from this simple argument, it is known in much greater generality that derivatives of order higher than two do not yield in general contraction semigroups on spaces L^p other than p =2, <cit.>. However, there is still a class of fourth order operators that generates contractive semigroups on L^p for p∈[3/2,3] <cit.>; as also proved in <cit.> in the case of the Friedrichs realizations of the bi-Laplacians on finite connected graphs through Nittka's characterization of L^p-contractivity.


We proceed with a study of positivity and the Feller property for semigroups:



A Banach lattice is a Banach space X such that for all f,g∈ X | f |≤| g | implies ‖ f ‖≤‖ g ‖.
Examples of such spaces include the spaces X_c = L^p(𝒢), C_0(𝒢), and BUC(𝒢). A C_0-semigroup of operators (T(t)) on a Banach lattice is then called positive if f≥ 0 implies that T(t) f ≥ 0 for all t ≥ 0. 



The Schrödinger semigroup with V ∈ L^∞ is positive on all spaces L^p with p ∈ [1, ∞) and spaces of continuous functions C_0(𝒢) and BUC(𝒢). In fact, the heat semigroup is even positive improving, i.e. a function f ≠ 0 such that f ≥ 0 gets mapped to a function (T(t)f)(x)>0 for all t>0 and x ∈𝒢.


We start with X_c=L^p(𝒢) and p< ∞.
Take f ∈ L^p(𝒢) positive and some f_n ∈ L^p(𝒢) ∩ L^2(𝒢) positive such that lim_n →∞‖ f_n -f ‖_L^p(𝒢)=0. Then from standard operator theory T_2(t)f_n ≥ 0 (Trotter–Kato theorem and Beurling-Deny criterion). On the other hand, lim_n →∞‖ T_1(t)f_n- T_1(t)f ‖_L^1(𝒢)=0 by continuity of the semigroup. 
Standard measure theory yields the existence of a pointwise a.e. convergent subsequence such that T_1(t)f ≥ 0.
Take f ∈ BUC(𝒢) positive and any g ∈ L^1(𝒢) positive, then the dual pairing is positive (T_1(t)g,f)_L^1(𝒢) × L^∞(𝒢)≥ 0.
This however also implies that for all g ∈ L^1(𝒢) positive (g,T_BUC(t)f)_L^1(𝒢) × L^∞(𝒢)≥ 0.
If T_BUC(t)f was not a positive function, then there was a small open interval I on some edge, where this function is negative. Taking g as the characteristic function of that interval g=χ_I, we would obtain that (g,T_BUC(t)f)_L^1(𝒢) × L^∞(𝒢)< 0
which contradicts positivity above.



Now, apart from this abstract argument for positivity, it can be directly seen from the heat kernel construction that the heat semigroup is positivity improving, as the heat kernel is strictly positive everywhere.
 

We immediately obtain the following corollary from Lemma <ref> and Prop. <ref>.


The free Schrödinger semigroup (T_C_0(t)) on C_0(𝒢) is a Feller semigroup. It also has the strong Feller property.


Although it is well-known that the biharmonic heat kernel is non-positive, we can still obtain the following positivity result for the biharmonic equation if the conductivities are summable


We consider an equilateral graph that satisfies c(𝒢)<∞. Then the semigroup e^-t(-Δ)^m is uniformly asymptotically positive on L^2(𝒢), i.e., for every ε>0 there is t_0 ≥ 0 such that 

    d (E_+,e^t(-(-Δ)^m- inf (σ(-(-Δ)^m)))f) ≤ε‖ f ‖ for all t ≥ t_0 and all f ∈ E_+,
 where E_+:={u ∈ L^2(𝒢); u ≥ 0 }. 



Using results from subsequent Sections (Prop. <ref>, Cor. <ref>, Theorem <ref>) it follows that the spectral bound inf (σ(-(-Δ)^m)) is isolated in the spectrum, and corresponds to a finite eigenvalue. Thus, <cit.> implies then the claim.



Let 𝒢 be a metric graph that satisfies Assumption <ref>. Schrödinger semigroups T_q(t)=e^-t(-Δ_q+V) with relatively (-Δ_q)-bounded potential, with relative bound <1, are analytic on spaces X_c=L^p and spaces C_0(𝒢), BUC(𝒢). Moreover, the higher order Laplacians -(-Δ_q)^m, with m ≥ 1, generate analytic semigroups on all these spaces as well. In the Hilbert space case L^2(𝒢), the previous results hold also without imposing Assumption <ref>.


On L^2(𝒢), self-adjointness of the generator implies immediately the claim.

On all other spaces, it suffices to analyze operators -(-Δ_q)^m, only. The analyticity of the Schrödinger operator then follows from perturbation theory, cf. <cit.>.

We start by showing that generators A_q, of semigroups (T_q(t)) are indeed given by -(-Δ_q)^m.
Let q≤ 2 first, then for integers j < m, and α:=(j-1)q+2/mq satisfying p^-1=1-q^-1=j - mα + q^-1, we find by Gagliardo–Nirenberg interpolation and the uniform boundedness of edge-lengths and conductivities from above and below that there exist universal constants C_1,C_2>0 such that 

    ‖ u^(j)‖_p≤ C_1 ‖ u^(m)‖_q^α‖ u ‖_q^1-α+ C_2 ‖ u ‖_q.

This implies that if there exists a solution -(-Δ_q)^m u = λ u, then 

    λ⟨ u , u ⟩_L^p × L^q= -⟨ (-Δ)^m_q u , u ⟩_L^p × L^q = -⟨ u^(m) , u^(m)⟩_L^p × L^q≤ 0,

which implies λ≤ 0. This implies that -(-Δ_q)^m-λ is injective for λ>0. Since the semigroup maps into D(-(-Δ_q)^m) it follows that the generator A_q satisfies A_q ⊂ -(-Δ_q)^m. Hence, for λ >0 large enough we have A_q-λ = -(-Δ_q)^m-λ such that -(-Δ_q)^m-λ is invertible, too. This implies also that operators -(-Δ_p)^m-λ with p ∈ [2,∞) as well as -(-Δ_C_0)^m-λ and -(-Δ_BUC)^m-λ are injective (duality). Hence, we conclude that the generators are indeed given by A_p= -(-Δ_p)^m, A_C_0= -(-Δ_C_0)^m, and A_BUC=-(-Δ_BUC)^m.

To show analyticity, recall that a closed, densely defined linear operator A_q generates an analytic semigroup if and only if there exists a half-plane (λ)>ω contained in ρ(A_q) such that ‖ (λ-A_q)^-1‖≲|λ-ω|^-1.
All the above operators are closed and densely defined by Lemma <ref>.

First, we can conclude from spectral equivalence, Corollary <ref>, that the resolvent set contains a sector and satisfies, by the Combes–Thomas estimate, Proposition <ref>, and consistency of resolvents, the estimate for A=-(-Δ)^m on L^p(𝒢) and (λ)>ω=ε

    ‖ (λ-A_q)^-1‖≲|λ-ω|^-1.


For spaces C_0(𝒢) and BUC(𝒢) it suffices to note that, using Prop. <ref>

    ‖ (λ-A)^-1‖_ℒ(C_0)≤‖ (λ-A)^-1‖_ℒ(BUC)   ≤‖ (λ-A)^-1‖_ℒ(L^∞)= ‖ (λ-A)^-1‖_ℒ(L^1).

 This concludes the proof.


Immediately from <cit.>, we obtain the following.


Schrödinger semigroups satisfy a spectral mapping theorem e^-t σ(H) = σ(T(t)) \{0},  t ≥ 0
and the spectral bound and growth bound of the semigroups T(t) coincide.






 §.§ Asymptotic properties


Recall that a C_0-semigroup (T(t)) on X_c is called strongly stable if for all x ∈X_c we have lim_t →∞‖ T(t)x ‖=0, uniformly stable if lim_t →∞‖ T(t) ‖=0,
and uniformly exponentially stable, if there is ε>0 such that lim_t →∞e^ε t‖ T(t) ‖=0.


From our previous discussions we immediately conclude that



Let Assumption <ref> be satisfied. The semigroup generated by -(-Δ)^m is never uniformly stable on any of the spaces X_c=L^p(𝒢), C_0(𝒢), BUC(𝒢). 


It follows from <cit.> that uniform stability is equivalent to exponential stability. To have exponential stability the generator needs to possess a spectral gap. This is however not the case as 0 ∈σ_p(Δ_BUC) which follows from Δ_BUC=0.

Because T(t)=, on the space of bounded uniformly continuous functions the heat semigroup is not strongly stable; on all other spaces it may or may not be so. 

If we drop the assumption of sub-exponential growth on our graph, the heat semigroup can indeed become exponentially stable: 

Let 𝒢 be the homogeneous equilateral tree of degree q ≥ 3 with unitary conductivities. The spectrum of Δ on L^2(𝒢) is given by <cit.>

    σ_p(Δ)    = {-k^2π^2; k ∈ℕ} and 
    σ_c(Δ)   ={- λ∈ (-∞,0]: cos(√(λ)) ∈[ -2√(q-1)/q,2√(q-1)/q] }.
 
In particular, Δ has a spectral gap at zero. This implies that for some M>0 and ω>0 we have exponential decay ‖ T(t) ‖≤ M e^-ω t. 






 


§ EQUILATERAL GRAPHS: POINT SPECTRA OF SCHRÖDINGER OPERATORS

In the following two sections, we assume the graph 𝒢 to be equilateral.

Since we assume the graph 𝒢 to be equilateral, this condition clearly implies that there is κ≥ 1 such that for all adjacent edges we have c(') ≤κ c()  if '∩≠∅.

In this section, we characterize the point spectrum of Schrödinger operators 

    Hψ:=-Δψ+Vψ,

with relatively -Δ bounded potential V=(V_) ∈ L^1_loc that is the same on every edge  with relative bound <1, on (<ref>).




Let H^D be the Schrödinger operator with Dirichlet boundary conditions

    D(H^D)   :={ f ∈ C^2([0,1]); f(0)=f(1)=0 },    H^D f:=Hf.

The spectrum of this operator is discrete.
We also define the operator H^D_X_c acting on X_c with Dirichlet boundary conditions on every edge.
There are always two linearly independent solutions c_λ,s_λ∈ C^∞[0,1] to the classical equation -Δψ+V_ψ= λψ
such that c_λ(0) =1, s_λ(0)=0 and c_λ'(0) =0, s_λ'(0)=1. 
For zero potential V ≡ 0, the solutions are explicitly given by 

    c_λ(t):=cos(√(λ)t) and s_λ(t):=sin(√(λ)t)/√(λ).



Let λ∉σ(H^D), a function satisfying H ψ_λ= λψ_λ can then be expressed in terms of s_λ and c_λ on edges  as

    ψ_λ (t) = ψ_λ(1)-ψ_λ(0)c_λ(1)/s_λ(1)s_λ(t)+ψ_λ(0) c_λ(t).

Consider the operator H^ext_X_c: D(H^ext_X_c) ⊂X_c→X_c defined by

    D(H^ext_X_c)   :={ 
    f ∈X_c :  f ∈ C(𝒢)  and H f ∈X_c}, H^ext_X_cψ:=Hψ.

We stress that no Kirchhoff conditions at 0,1 are imposed on the elements of D(H^ext_X_c).



As is well-known in the Hilbert space setting, there is a close connection between the discrete Laplacian on the underlying lattice of the graph and the continuous Laplacian on the metric graph. Our aim is to extend this idea to the function spaces we study in this article. 

To extend this idea to also other spaces, we want to introduce the following correspondences between functions spaces on combinatorial (the vertex set) and metric graphs, 𝒢 that we use frequently throughout the article.






We then have the following Lemma linking spaces on vertex sets to spaces on the metric graph:



The vertex map π : D(H^ext_X_c) →X_d defined by π(ψ)():=ψ() restricts to an
isomorphism 

    π|_ker(H^ext_X_c-λ):ker(H^ext_X_c-λ) →X_d
 for any λ∈ρ(H^D).



Its inverse is the gamma field γ_X_d: ρ(H^D) →ℒ(X_d,X_c^(2)), with X_c^(2) defined in (<ref>), satisfying 

    π|_ker(H^ext_X_c-λ)∘γ_X_d(λ)=id_X_d and γ_X_d(λ) ∘π|_ker(H^ext_X_c-λ)=id_ker(H^ext_X_c-λ)
 on edges ∈ℰ(𝒢) and is given by

    (γ_X_d(λ)z)_(t)= s_λ(t)z(t())+z(i())(s_λ(1)c_λ(t)-c_λ(1)s_λ(t))/s_λ(1).



The algebraic relations (<ref>) are immediate.
Also, the functions (γ_X_d(λ)z) and H_X_c(γ_X_d(λ)z) are in X_c since functions c_λ and s_λ are smooth. 
Using that c_λ satisfies H_X_cc_λ=λ c_λ, and similarly for s_λ, this implies 

    ‖γ_X_d(λ)z ‖_X_c^(2)   = ‖ H_X_cγ_X_d(λ)z ‖_X_c + ‖γ_X_d(λ)z ‖_X_c
       ≲ (|λ|+1) ‖γ_X_d(λ)z ‖_X_c≲_λ‖ z ‖_X_d

since on every edge  there is a constant C_λ>0 independent of  such that 

    | (γ_X_d(λ)z)_(t) |≤ C_λ (| z(i()) |+ | z(t()) |).

Since the two maps t ↦s_λ(t)s_λ(1) and t ↦s_λ(1)c_λ(t)-c_λ(1)s_λ(t)/s_λ(1) are continuous and vanish / attain the value one at t=0 and t=1, respectively, we have that for some sufficiently small ε>0, independent of ,

    | z(i()) |≤ C_λ,p,ε‖ (γ_X_d(λ)z)_‖_L^p(0,ε) and | z(t()) |≤ C_λ,p,ε‖ (γ_X_d(λ)z)_‖_L^p(1-ε,1).

From this, we get that γ_X_d(λ) ∈ℒ(X_d,X_c^(2)).

We also consider the map μ_X_d∈ℒ(X_c,X_d) defined as 

    (μ_X_d(λ)f)(i())   = ⟨ (s_λ(1)c_λ-c_λ(1)s_λ),f_⟩/s_λ(1) and  
    (μ_X_d(λ)f)(t())= ⟨ s_λ,f_⟩/s_λ(1).

In the Hilbert space case X_c=L^2(𝒢) and X_d=ℓ^2(𝒱), we have μ_ℓ^2(λ)=γ_ℓ^2(λ)^*. 

The next Lemma provides us with an isomorphism, expressed in terms of the Floquet discriminant, <cit.>, D(λ):=12 (c_λ(1)+s_λ'(1)) on ker(H_X_c-λ) rather than ker(H^ext_X_c-λ)⊃ker(H_X_c-λ).


For λ∈ρ(H^D_X_c) the operator M(λ)_X_d∈ℒ(X_d), with Floquet discriminant D(λ):=12 (c_λ(1)+s_λ'(1)), is given by

    (M(λ)_X_dz)()= ∑_∈ℰ_(𝒢)c()(z(t()1_∈ i(ℰ(𝒢))+i()1_∈ t(ℰ(𝒢)))-D(λ)/s_λ(1))

and has the property that γ_X_d(λ)|_ker(M(λ)_X_d): ker(M(λ)_X_d) →ker(H_X_c-λ) 
is an isomorphism.





We only need to check the Kirchhoff condition on the derivatives. 
We first observe that the Wronskian W(s_λ,c_λ)(t):=s_λ'(t)c_λ(t)-s_λ(t)c_λ'(t)
satisfies W(s_λ,c_λ)(0)=1 from the initial conditions and since the Wronskian is constant we have W(s_λ,c_λ)(t)=1  for all t ∈ [0,1].



This allows us to introduce the Dirichlet-to-Neumann map by differentiating (<ref>)

    m(λ) := 1/s_λ(1)(  -c_λ(1)     1 
     1     -s_λ'(1) )

such that

    ( (γ_X_d(λ)z)'_(0) 
     -(γ_X_d(λ)z)'_(1) ) = m(λ) (  z_(0) 
     z_(1) ).

We therefore obtain for the derivatives at vertices ∈𝒱(𝒢)

    0=∑_∈ℰ_(𝒢)∂ (γ_X_d(λ)z)_/∂ n_()=(M(λ)_X_dz)().

This implies immediately that 0 ∈σ_p(M(λ)_X_d) iff λ∈σ_p(H_X_c).

By rewriting the operator M(λ)_X_d we obtain the following corollary.


For λ∈ρ(H^D) we have that λ∈σ_p(H_X_c) if and only if the Floquet discriminant D(λ), satisfies D(λ) ∈σ_p(P_X_d) where 

    (P_X_dz)():= ∑_∈ℰ_(𝒢)c()/c() z(t()1_∈ i(ℰ(𝒢))+i() 1_∈ t(ℰ(𝒢)))  on X_d.



We then have that 


If c(𝒢) < ∞, then the discrete Laplacian is a compact operator on all spaces ℓ^p(𝒱(𝒢)) with p ∈ (1,∞).


The following estimate shows the compactness on ℓ^p with 1/p+1/q=1 and p∈ (1,∞), for which we fix some vertex o ∈𝒱,

    ‖_B(o,r)^c P z ‖^p_p = ∑_∈𝒱(𝒢\ B(o,r)) c()^1-p|∑_∈ℰ_(𝒢)c() z(t()1_∈ i(ℰ(𝒢))+i() 1_∈ t(ℰ(𝒢))) |^p 
       ≤∑_∈𝒱(𝒢\ B(o,r)) c()^1-p( ∑_;i()=c() | z(t()) |^p +∑_;t()=c() | z(i()) |^p)( ∑_∈ℰ_(𝒢)c()^q )^p/q
       ≤∑_∈𝒱(𝒢\ B(o,r)) c() ( ∑_;i()=c() | z(t()) |^p +∑_;t()=c() | z(i()) |^p) 0.

Hence, the operator P is approximated by finite-rank operators _B(o,r) P which implies the claim. 
This concludes the proof.

This condition however, is not sufficient to obtain compactness on c_0, ℓ^1, and ℓ^∞ as the following example shows:

To see that the operator is not compact on ℓ^∞(), and thus by Schauder's theorem also on c_0 and ℓ^1, let us choose conductivities c(n,n+1)=1/| n |^2+1. Then 

    c(n,n+1)/c(n) = n^2-2n+2/2n^2-2n+3.

It is straightforward to verify that with this choice of conductivities P is not compact, since the image of the standard basis (e_n)_n ∈ℕ under P does not have a convergent subsequence.


It follows from ℓ^p ⊂ℓ^2, assuming still that c(𝒢)< ∞, for p ∈ (2,∞) from standard properties, see e.g. <cit.>, that (P_ℓ^2-λ)= (P_ℓ^p-λ) for all λ∈ and p∈ (2,∞).





Let us now state a lower growth bound on solutions φ to the equation H_X_cφ = λφ for λ such that D(λ) is non-real under a suitable growth condition on the graph.

Fix an element o ∈𝒱(𝒢) and consider a nested partition 𝒢_n:={ x ∈𝒢; | d(x,o) |≤ n } such that ⋃_n𝒢_n = 𝒢. Moreover, assume that the graph satisfies the growth condition

    lim sup_n →∞|𝒱(𝒢_n) |/ c^n =0

for any c>1. Then, the point spectrum of H_X_c satisfies σ_p(H_X_c) ⊂{λ∈ℂ; D(λ) ∈ℝ}. In particular, the ℓ^2 mass of any solution in X_d to the above equation satisfies for some c>0 that ⟨_𝒢_n z,z ⟩_ℓ^2(𝒱(𝒢))≥ c r^n, where r is the unique solution in (1,∞) to the equation r^2-1= r |(D(λ)) |/d_v^*.


Let λ∈ℂ\ℝ and assume there was an eigenfunction φ∈ D(Δ_X_c) such that

    (-Δ_X_c+V) φ=λφ.
 
This implies by Corollary (<ref>) the existence of an eigenfunction z ∈X_d such that

    P_X_dz = D(λ) z.

We therefore conclude that

    S_n   :=⟨_𝒢_nP_X_dz,z ⟩_ℓ^2(𝒱(𝒢))
    =∑_v ∈𝒱(𝒢_n)c()(P_X_dz)() z()
       =⟨_𝒢_nD(λ)z,z ⟩_ℓ^2(𝒱(𝒢))=D(λ) ∑_v ∈𝒱(𝒢_n) c()| z() |^2.

We observe that any edge that is only partially contained in 𝒢_n is fully contained in Ω_n+1.
Taking the absolute value of the imaginary part, we obtain the estimate 

    | S_n |   = |(D(λ)) |∑_v ∈𝒱(𝒢_n) c() | z() |^2 
       =|(∑_∈𝒱(𝒢_n)c()(P_X_dz)() z())|≤ d_^* ∑_∈𝒱(𝒢_n+1\𝒢_n-1)c() | z() |^2

which by introducing s_n:=⟨_𝒢_n z,z ⟩_ℓ^2(𝒱(𝒢))>0 means that s_n+1-s_n-1≥|(D(λ)) |/d_v^* s_n.
Without loss of generality we may assume s_1 >0 otherwise we translate the origin of the graph.
Then, since all s_n>0 we obtain a lower bound on s_n by positive δ_n satisfying δ_n+1-δ_n-1= |(D(λ)) |/d_v^*δ_n with δ_1:=s_1.
Solutions to this three term recurrence relation are, in terms of two constants λ_1, λ_2 ∈ℝ, given by δ_n = λ_1 r_1^n+ λ_2 r_2^n, where r_1 ∈ (-1,0) and r_2 ∈ (1, ∞) are solutions to the equation x^2-1= |(Δ(λ)) |/d_v^* x. In particular λ_2 is strictly positive.
Thus, for large n we see that r_2^n≲ s_n where r_2>1.
This contradicts the boundedness of eigenfunctions in all spaces that we consider. 

Let us record an immediate fact about the spectrum.


0 ∈σ_p(Δ_BUC) is a simple eigenvalue for any infinite graph.


Δ_BUC 1 = 0. To prove the second assertion, observe that each element of the null space of Δ_BUC is necessarily a piecewise affine function, and indeed a constant function on each connected component, as one sees enforcing boundedness. The claim now follows in view of our standing assumption of connectedness of 𝒢.






Results on the Dirichlet spectrum highly depend on geometric properties of the graph results for various cases can be found in <cit.>. In addition, we have the following immediate result:

If the graph 𝒢 is an equilateral tree, then the Dirichlet spectrum belongs to σ_p(Δ_BUC).


Since functions e^± i λ x are bounded, this just follows from iteratively adding new fragments of such functions, with the correct boundary conditions, along the tree.





§ EQUILATERAL GRAPHS: CONTINUOUS SPECTRUM OF THE KIRCHHOFF-LAPLACIAN


In this section we are still assuming 𝒢 to be equilateral and make the following growth assumption: 
[Growth condition-Equilateral graphs]

We assume that 𝒢 is an equilateral graph such that for all ε>0 there is C_ε>0 such that for all ∈ℰ(𝒢) 

    c(B_r(x) ⊃ℰ( 𝒢)) ≤ C_ε e^ε r c(),    x=i(),

where B_r(x) ⊃ℰ( 𝒢) is the set of edges fully contained in B_r(x). Here, we write for A ⊂ℰ(𝒢) the conductivity c(A) := ∑_∈ A c().




Given λ∈ρ(Δ^D) and g=(g_) ∈X_c, we introduce the function u ∈ D(Δ_X_c) defined on edges ∈ℰ(𝒢) by

    u_(x):=   s_λ(x)/s_λ(1)(w(t())- ∫_x^1 g_(y) s_λ(1-y)  dy) 
       +s_λ(1-x)/s_λ(1)(w(i())-∫_0^x g_(y)s_λ(y)  dy ),

where s_λ is defined as in (<ref>);
u satisfies the equation (Δ_X_c-λ)u=g
for any family of coefficients (w())_∈𝒱(𝒢) in X_d.

Consider then functions f_1,λ(x):=s_λ(1-x) and f_2,λ(x):=s_λ(x).

These two functions are linearly independent on (0,1) for λ∉ρ(Δ^D) since 

    f_1,λ(0)=s_λ(1)≠ 0=s_λ(0)=f_2,λ(0).




There exists a function g ∈ C[0,1] such that g(0)=g(1)=0 and

    ⟨ g_λ,f_1,λ⟩_L^2(0,1) =0 and ⟨ g_λ,f_2,λ⟩_L^2(0,1) =1.



Consider the measure dμ(x) = x(1-x)  dx on (0,1).
There is by Gram–Schmidt orthonormalization a function g_λ(x):=g_λ(x)x(1-x) such that ⟨g_λ,f_1,λ⟩_L^2(dμ) =0 and ⟨ g_λ,f_2,λ⟩_L^2(dμ) =1. If g would not vanish at the end-points e.g. at x=0, then |g_λ(x)|≥ c/x for some c>0 and thus g_λ∉ L^2((0,1),dμ).






Let W_λ∈X_d be given, then there is a function y ∈X_c, that vanishes at the vertices, such that 

    W_λ()= ∑_∈ℰ_c()/c()∫_0^1 y_(x)f_I(),λ(x)  dx

where I()=1 if i()= and I()=2 otherwise.


Let T be a spanning tree of 𝒢, i.e., a new metric graph that shares all edges with 𝒢 but such that some of the vertices can be cut through, cf. <cit.>, and let it be oriented in such a way that for a given root o of the tree an edge 


  * with a vertex  of degree one, is oriented such that i():=; 

  * with vertices of degree ≥ 2, is oriented such that we choose the initial vertex to be the vertex with smaller geodesic distance to o.

This way, there is for every ∈𝒱(𝒢) some ∈ℰ(𝒢) such that i()=.
We then set y_(x):=α_,λ g_λ(x) with g_λ as defined in Lemma <ref>.
By Assumption <ref> we can define the positive quantity c_0():=∑_∈ℰ(T): i()=c() such that c()/c_0() is uniformly bounded. Then setting α_,λ:=c()/c_0() W_λ() if ∈ℰ(T) with =i() and α_,λ=0 if ∉ℰ (T) provides the desired function y in X_c. That y is indeed a function in X_c follows since for p <∞

    ‖ y ‖_p≤( ∑_∈ℰ(𝒢)|α_,λ|^p ‖ g_λ‖^p_p)^1/p < ∞

and analogously for p=∞.


We have established the following correspondence between the spectra of the discrete Laplacian and continuous Laplacian on metric graphs:



Let 𝒢 be an equilateral metric graph that satisfies Assumption <ref>. 
The spectrum of Δ_X_c on X_c∈{ L^p(𝒢), C_0(𝒢), BUC(𝒢)} away from the point spectrum σ_p(Δ_X_c) and the Dirichlet spectrum σ(Δ^D) coincides with 

    {λ∉ (σ_p(Δ_X_c)∪σ(Δ^D)); c_λ(1) ∈σ(P_X_d) },

where the correspondence between X_c and X_d follows the scheme in Table <ref>.


We assume throughout the proof that λ∉ (σ_p(Δ_X_c)∪σ(Δ^D)).

Given arbitrary c_λ(1) ∈ρ(P_X_d) and g ∈X_c, we have to construct u ∈ D(Δ_X_c) such that (Δ_X_c -λ )u=g.

Using g, we obtain the sequence (W_λ())_∈𝒱(𝒢)∈X_d, as in Lemma <ref>, by the formula

    W_λ()= ∑_∈ℰ_c()/c()∫_0^1 g_(x)f_I(),λ(x)  dx.

By assumption there is U ∈X_d such that 

    (P_X_d-c_λ(1))U=W_λ.

Then, by inserting the respective values of U into

    u_(x):=   s_λ(x)/s_λ(1)(U(t())- ∫_x^1 g_a(y) s_λ(1-y)  dy) 
       +s_λ(1-x)/s_λ(1)(U(i())-∫_0^x g_a(y)s_λ(y)  dy )

we obtain the desired function that solves (Δ_X_c -λ )u=g. 
A direct computation and (<ref>) show that the Kirchhoff conditions are satisfied as well

    (M(λ)_X_du)()
       =∑_∈ℰ_(𝒢)∂ u_/∂ n_() = c()/s_λ(1)(-(P_X_d-c_λ(1))U+W_λ) = 0.

For the converse implication, assume that λ∈ρ(Δ_X_c) and W ∈X_d. By the previous Lemma <ref> there is y ∈X_c such that 

    W_λ()= ∑_e ∈ℰ_c()/c()∫_0^1 y_(x)f_I(),λ(x)  dx.

Then, there is u ∈ D(X_c), i.e. u satisfies in addition the Kirchhoff condition, such that (Δ_X_c-λ)u=y. Let U ∈X_d be the vertex points associated with u, then the computation in (<ref>) shows that (P_X_d-c_λ(1))U=W_λ.
This completes the proof.


From our discussion so far, we conclude the following spectral independence result for equilateral graphs satisfying the sub-exponential growth condition in Assumption <ref>:
[Spectral independence of Laplacian]

The spectrum of the Laplacian Δ_X_c, for λ∈ρ(H^D) on equilateral graphs satisfying Assumption <ref>, is independent of the space X_c∈{ L^p(𝒢), C_0(𝒢), BUC(𝒢) }, i.e., σ(Δ_X_c) ∩ρ(H^D)=σ(Δ_L^2) ∩ρ(H^D).


It suffices to observe that the operator P_X_d, defined in (<ref>), is of the form considered in <cit.>.
This result follows by combining Theorem <ref> with <cit.>. For the space c_0, which is not discussed in <cit.>, we use that c_0^**=ℓ^∞.





As a consequence of the consistency of the resolvents, cf. (<ref>), we show that the Kreĭn resolvent formula extends to spaces X_c = L^p(𝒢) with p < ∞:
[Kreĭn resolvent formula] Let Assumption <ref> be satisfied and 𝒢 be equilateral. We consider the Schrödinger operator H_p=-Δ+V with potential V ∈ L^∞(𝒢), that is the same on every edge, on L^p(𝒢). Let z ∈ρ(H^D) ∩ρ(H_p), where H^D is the Schrödinger operator with Dirichlet boundary conditions on every edge. Then the operator M_ℓ^p, defined in (<ref>), is invertible and satisfies (H_p-z)^-1 = (H_p^D-z)^-1+ γ_ℓ^p(z)M_ℓ^p(z)^-1μ_ℓ^p(z).
Here γ_ℓ^p is the gamma field introduced in (<ref>).


Since this formula is well-known to hold on L^2(𝒢), see e.g. <cit.> and references therein, and we have already established spectral independence in Theorem <ref> and consistency of resolvents in (<ref>), we can use an L^2-approximation argument: 
If we then take a sequence u_n ∈ C_c(𝒢) approximating some u ∈X_c, the following limits exist:

    (H_p-z)^-1u = lim_n →∞, ‖∙‖_X_c (H_2-z)^-1u_n, 
       (H^D_p-z)^-1u = lim_n →∞, ‖∙‖_X_c (H^D_2-z)^-1u_n ,  and 
       (γ_ℓ^p(z)M_ℓ^p(z)^-1μ_ℓ^p(z)) u = lim_n →∞, ‖∙‖_ℓ^p(γ_ℓ^2(z)M_ℓ^2(z)^-1γ_ℓ^2(z)^*) u_n.

This concludes the proof.






§ MARKOV SEMIGROUP AND BROWNIAN MOTION ON METRIC GRAPHS


Based on our preliminary work in previous sections, we are now in the position to introduce a canonical stochastic process which we will refer to as Brownian motion on infinite metric graphs, cf. <cit.> for related results on finite graphs. 

[Brownian motion]
We define the Brownian motion (B_t) on a metric graph as the Feller process associated with the rescaled heat semigroup A_BM := 1/2Δ on BUC(𝒢).


The theory of Feller processes gives immediately the Feynman–Kac formula <cit.>. An analogous formula for discrete graphs has been proved in <cit.>.


Consider the Feller process B_t with generator Δ_C_0 and a function V ∈ C(𝒢).
We then define for f ∈ D(Δ_C_0)

    u_f(t,x) =𝔼^x ( f(X_t)exp( ∫_0^t -V(X_s)  ds )).

It follows that u_f(t,x) ∈ D(Δ_C_0) and u_f solves the parabolic Schrödinger equation

    ∂_t u_f(t,x)    = -(-Δ +V)u_f(t,x),    t>0 
    
    u_f(0,x)   =f.


We conclude with an immediate corollary on heat kernel estimates for Schrödinger semigroups and higher-order parabolic equations.

[Heat kernel estimates]

Let 𝒢 be a metric graph and V ∈ C^∞(ℰ(𝒢)) such that all its derivatives are bounded over the entire metric graph.
 Then the heat kernel of the Schrödinger semigroups and higher-order parabolic equations satisfies k_t ∈ C^∞(ℰ(𝒢) ×ℰ(𝒢)). 

For potentials V ∈ C(𝒢) ∩ L^∞(𝒢) and all x,y ∈𝒢, where 𝒢 satisfies Assumption <ref>, we have for all t >0

    | e^-tH(x,y) |≲| e^tΔ(x,y) |^1/2 t^-1/4 e^t ‖ V^+ ‖_∞

where V^+ is the positive part of V.
Moreover, for semigroups | e^-t(-Δ)^m| there is C_t_0>0 such that for all x,y ∈𝒢 and t ≥ t_0 

    | e^-t(-Δ)^m(x,y)|≤ C_t_0 t^-1/(2m)exp(-c_22t^1/(2m-1)| d(x,y) |^2m/2m-1 + c_3 t)

where c_2 and c_3 are as in (<ref>).


The functional calculus implies that for t>0 the semigroup T(t)=e^-tH is a map T(t) ∈ℒ( L^2(𝒢), H^m(𝒢)) for all m ≥ 0. Duality implies then that T(t) ∈ℒ(H^-n(𝒢),L^2(𝒢)) for n ≥ 0 such that by the semigroup property

    T(t+s) ∈ℒ(H^-n(𝒢),H^m(𝒢))  for all m,n ∈ℕ_0.

This implies that T(t) ∈ℒ(ℰ'(ℰ(𝒢)),ℰ(ℰ(𝒢))) for all t>0.



The existence of the integral kernel representation for the Schrödinger semigroup follows by Schwartz's kernel theorem <cit.>. Since the semigroup is smoothing for t>0 we conclude that the integral kernel must be given by a smooth function <cit.>. 
The p-independence of the spectrum of the semigroup (away from zero) follows from the spectral mapping theorem in Corollary <ref>.




Hölder's inequality in the Feynman–Kac formula implies that 

    | u_f(t,x) |   ≤𝔼^x ( | f(B_t) |^p^-1+q^-1exp( ∫_0^t -V(B_s)  ds )) 
       ≤𝔼^x ( | f(B_t) |)^q^-1𝔼^x ( | f(B_t) |exp( ∫_0^t -pV(B_s)  ds ))^p^-1
       = (e^tΔ| f |(x))^1/q(e^t(Δ-pV)| f |(x))^1/p.

If we now choose f to be an approximate identity 0 ≤ f_n →δ_y we find using the ultracontractivity of the Schrödinger semigroup, i.e., Theorem <ref>, that

    | e^-tH(x,y) |   ≤| e^tΔ(x,y) |^1/q| e^t(Δ-pV)(x,y) |^1/p
       ≤| e^tΔ(x,y) |^1/qsup_x',y'| e^t(Δ-pV)(x',y') |^1/p
       = | e^tΔ(x,y) |^1/q‖ e^t(Δ-pV)‖_ℒ(L^1,L^∞)^1/p
       ≲| e^tΔ(x,y) |^1/q t^-1/(2p) e^t ‖ V^+ ‖_∞


The kernel estimates on the graph then follow by using that for d(x,y) ≥ n ℓ_↓ and k_t the integral kernel of the semigroup (e^-t(-Δ)^m)_t ≥ 0 on , we find from kernel estimates (<ref>), estimate (<ref>) on the transfer coefficients, and the definition of the kernel on the graph (<ref>)

    | K_k_t(x,y) |   ≲ t^-1/(2m)exp(-c_2 | d(x,y) |^2m/2m-12t^1/(2m-1)+ c_3 t) ∑_k ≥ nexp(-c_22t^1/(2m-1)| k ℓ_↓|^2m/2m-1)3^k 
       ≲ t^-1/(2m)exp(-c_22t^1/(2m-1)| d(x,y) |^2m/2m-1 + c_3 t).

Here, we used that ∑_k ≥ nexp(-c_22t^1/(2m-1)| k ℓ_↓|^2m/2m-1)3^k is uniformly bounded for t ≥ t_0 in n.

















The process (B_t) has a.s. continuous paths and has martingales M_t:=B_t and M_t:=| B_t |^2-t on the cubic metric graph 𝒢 induced by the integer lattice 𝒱:=^d with unit conductivities. 


The continuity of paths follows by the locality of the graph Laplacian <cit.>. 

The martingales can be derived directly from the Markov generator using Dynkin's formula: 

Let f ∈ D(Δ_C_0) then ℳ_t^f:= f(B_t) - f(B_0) - ∫_0^t 1/2Δ f(B_s)  ds
defines a martingale with respect to the filtration induced by the process (B_t)_t ≥ 0.

The above martingales can be derived directly by approximating the (unbounded) functions f:𝒢→𝒢 with f(x) := x and f:𝒢→ with f(x):=| x|^2 on sufficiently large sets of growing size.

More precisely, let f_n(x) = χ_n(x) f(x) ∈ D(Δ_C_0), where χ_n is a smooth cut-off function of modulus at most one, such that χ_n(x)=1 for | x |≤ n and χ_n(x) = 0 for | x |≥ 2n.

We then have that from the kernel estimates in Theorem <ref>

    𝔼^x( (f -f_n)(B_t) )   = T_t (f (1- χ_n))(x) 
       = ∫_B_n(0)^c e^tΔ(x,y)f(y) (1-χ_n(y))  dy 0.


This implies that lim_n →∞𝔼^x | M_t^f_n- M_t^f| =0  for all  t>0 which shows that M_t^t is a martingale as well, even though f ∉ D(Δ_C_0).






§ PARABOLIC ANDERSON MODEL


We next turn to an application of our semigroup framework and introduce the parabolic Anderson model (PAM) on a periodic metric graph as the Cauchy problem

    ∂_t u(t,x)    =-(- Δ + V_ω) u(t,x), (t,x) ∈ (0,∞) ×𝒢 such that u(0,x) = .


This model can be seen as a natural extension of the discrete parabolic Anderson model on combinatorial graphs <cit.>. 
Here, we assume that V_ω are i.i.d.  realizations of a potential on every edge ∈ℰ(𝒢). From our analysis so far, we know that there exists a random semigroup (ω↦ (T_ω(t))_t≥ 0) associated with the evolution equation (<ref>) for BUC(𝒢) initial data.



In the following, we assume that 𝒢 is a periodic equilateral metric graph with fundamental domain D.


We can now prove a version of intermittency for our random Schrödinger operator using the tools from semigroup theory we developed in the preceding chapters.
For this purpose, we define functions Λ_p(t):= log𝔼∫_D u(t,x)^p  dx/| D | for t ≥ 0, p ∈ℕ.

Since the function p ↦Λ_p(t) is convex, difference quotients p ↦Λ_p+1(t) -Λ_p(t) are increasing in p.

Hence, we conclude that for q ≥ p

    qΛ_q+1(t) - (q+1)Λ_q(t) 
       = ∑_k=0^q-1((Λ_q+1(t)-Λ_q(t))-(Λ_k+1(t)-Λ_k(t)) ) 
       ≥∑_k=0^p-1((Λ_p+1(t)-Λ_p(t))-(Λ_k+1(t)-Λ_k(t)) ) 
       =pΛ_p+1(t)-(p+1) Λ_p(t).


We then write f ≪ g to denote that lim_t →∞ g(t)-f(t) = ∞.
Thus, if (p+1) Λ_p ≪ pΛ_p+1 then this implies also that (q+1) Λ_q(t) ≪ qΛ_q+1  for all  q ≥ p.



We also define translations (D_n)_n ∈ℤ^d, with D_0=D, of the fundamental domain such that 𝒢 = ⊔_n D_n.
[Intermittency]
The solution to the parabolic Anderson model is called intermittent if the Lyapunov exponents 

    λ_p:=lim_t →∞Λ_p(t)/t,  for  p ∈ℕ exist and we have λ_1 < λ_2/2 < λ_3/3 <...  .





Suppose H=-Δ+V such that V=(V_)_ is an i.i.d. family of bounded non-degenerate random potentials, then the parabolic Anderson model is intermittent.


Let k_ω(t,x,y) be the Schwartz kernel of the PAM.
We then have that since the probability measure is translation invariant

    𝔼∫_D u_(t+s,x)  dx/| D |   = 𝔼∫_D ∫_𝒢∫_𝒢 k_ω(s,x,y) k_ω(t,y,z)  dy  dz  dx/| D |
       = 𝔼∫_D ∑_n,m ∈ℤ^d∫_D_n∫_D_m k_ω(s,y,x) k_ω(t,y,z)  dy  dz  dx/| D |
       = 𝔼∫_D∑_n,m ∈ℤ^d∫_D_-m∫_D_n-m k_ω(s,y,x) k_ω(t,y,z)  dz  dx  dy/| D |
       = 𝔼∫_D ( u_(s,x) u_(t,x) )  dx/| D |.


Let λ_0:=supσ(-H_ω) a.s.. Then, we conclude from (<ref>) that 

    Λ_1(2t)=log𝔼∫_D u_(2t,x)  dx/| D | = log𝔼∫_D u_(t,x)^2  dx/| D |= Λ_2(t)

and using the Cauchy–Schwarz inequality 

    (𝔼∫_D u_(s+t,x)  dx/| D |)^2 ≤(𝔼∫_D u_(2s,x)  dx/| D |) (𝔼∫_D u_(2t,x)  dx/| D |).

Taking the logarithm of the previous line we find that 

    2Λ_1(s+t) ≤Λ_1(2s)+ Λ_1(2t)  for all  s,t ≥ 0.

Since Λ_1 is also continuous and satisfies Λ_1(s+t/2) ≤Λ_1(s)+ Λ_1(t)/2,
this implies the convexity of Λ_1.



Positivity of the semigroup yields that using the spectral theorem with spectral measure ℰ_H_ω for the operator H_ω

    𝔼∫_D u_(t,x)  dx/| D |≥𝔼⟨ T(t) _D, _D ⟩_L^2/| D |= 1/| D |𝔼∫_σ(H_ω)e^-λ t d⟨ℰ_H_ω(λ) _D, _D ⟩_L^2.

We have for any λ>λ_0 

    𝔼⟨ℰ_H_ω([λ_0,λ]) _D_n, _D_n⟩_L^2/| D |   =𝔼⟨ℰ_H_T^n ω([λ_0,λ]) _D_0, _D_0⟩_L^2/| D |
       =𝔼⟨ℰ_H_ω([λ_0,λ])_D,_D⟩_L^2/| D | >0.

This implies, by applying the logarithm to (<ref>), that lim_t →∞Λ_1(t)/t≥λ_0.



To show that lim_t →∞Λ_1(t)/t=λ_0 we use that the spectral bound coincides with the growth bound (Corollary <ref>)

    𝔼∫_D u_(t,x)  dx/| D | = 𝔼∫_D (T_ω(t) )(x)  dx/| D |≲ e^λ_0 t a.s.




Convexity implies then that for t_2 ≥ t_1

    ϕ(t_2):=Λ_1(t_2)/t_2=Λ_1(t_2)-Λ_1(0)/t_2≥Λ_1(t_1)-Λ_1(0)/t_1=Λ_1(t_1)/t_1=ϕ(t_1).


Thus, if we assume that λ=0 such that lim_t →∞Λ_1(t)/t=0, then by the monotonicity of Λ_1(t)/t we conclude that for all t>0 we have that Λ_1(t)/t ≤ 0. This implies Λ_1(t) ≤ 0.


Moreover, Λ_1 is non-increasing since from convexity we have for t_0<t_1<t_2

    Λ_1(t_2) ≥Λ_1(t_1)-Λ_1(t_0)/t_1-t_0 (t_2-t_1)+Λ_1(t_1).

Assuming that Λ_1(t_1)>Λ_1(t_0) for some 0<t_0<t_1 then the difference quotient in the previous line is positive.
Thus, we find that 

    lim sup_t_2 →∞Λ_1(t_2) ≥lim sup_t_2 →∞Λ_1(t_1)-Λ_1(t_0)/t_1-t_0 (t_2-t_1)+Λ_1(t_1)=∞

which is impossible as Λ_1(t) ≤ 0. Therefore, Λ_1(t) is non-increasing.
Hence, since Λ_1(t) is non-negative, we have the existence of some ρ such that lim_t →∞Λ_1(t)=ρ.
Thus, if ρ>-∞ we find that lim_t →∞Λ_2(t)-2Λ_1(t)= lim_t →∞Λ_1(2t)-2Λ_1(t)=-ρ.

On the other hand, if we assume that ρ=-∞, we may note that the function Λ_2(t)-2Λ_1(t) = Λ_1(2t)-2Λ_1(t)≥ 0 by convexity and is non-decreasing, as the difference quotient of convex functions is non-decreasing in both parameters.
This implies that there is c∈ [0,∞] such that lim_t →∞Λ_2(t)-2Λ_1(t) =c.

To show that in this case c=∞ we observe that 

    Λ_1(2^n t) -2^nΛ_1(t) = ∑_i=1^n 2^n-i(Λ_1(2^i t)-2 Λ_1(2^i-1 t)) ≤∑_i=1^n 2^n-ic = (2^n-1)c.

Thus, we have from dividing by 2^n t and taking the limit n →∞

    0= lim_n →∞Λ_1(2^n t) /2^n t≤Λ_1(t)+c/t  ∀ t≥ 0.

Now if c was not infinite, then choosing t large enough gives a contradiction since lim_t →∞Λ_1(t)=-∞.

From (<ref>) we then deduce that the random variable 

    Z_ω(x):=𝔼^x_(X_t) e^∫_0^∞-V_ω(X_s)+λ_0  ds∈ [0,1)

which is strictly less than one, as V_ω is non-degenerate describes the t →∞ limit of the solution by the Feynman–Kac formula, cf. Prop. <ref>. 
Moreover, by the dominated convergence theorem, we find 

    𝔼∫_D Z(x)  dx/| D |   = lim sup_t →∞e^λ_0 t𝔼∫_D (u_(t,x))  dx/| D |
       = lim sup_t →∞𝔼_V_ω∫_D 𝔼^x_(X_t) e^∫_0^t-V_ω(X_s)+λ_0  dsdx/| D |
       = lim sup_t →∞𝔼_V_ω∫_D (𝔼^x_(X_t) e^∫_0^t-V_ω(X_s)+λ_0  ds)^2 dx/| D | =𝔼∫_D Z(x)^2 dx/| D |.

Thus, since Z_ω(x) ∈ [0,1) it follows that Z_ω=0 a.s., which implies ρ=-∞ and the claim follows as c=∞.












§ COMBES–THOMAS ESTIMATE

For the Combes–Thomas estimate we introduce on edges  functions for a fixed reference vertex 

    (f_)_ (t):= d(i()+t,) + k_i(),χ_i(),(t)+ k_t(),χ_t(),(t)
 
where χ_i(),(0)=χ_t(),(||)=0 and (χ_i(),) is contained in a small neighborhood of 0 while (χ_t(),) is contained in a small neighborhood of ||. This implies that 

    (f_')_ (t)    =1+ k_i(),χ'_i(),(t)+ k_t(),χ'_t(),(t)  and 
     (f_^(n))_(t)    = k_i(),χ^(n)_i(),(t)+ k_t(),χ^(n)_t(),(t)  for  n ≥ 2

such that for some fixed ∈𝒱 we have both f_(0)=d(i(),) and f_(||)=d(t(),)
and coefficients k above are chosen such that the first derivative satisfies the natural boundary conditions at the vertices. Moreover, we assume that the first and second derivatives are uniformly bounded over the entire edge. This is possible as both the edge length and the conductivities are uniformly bounded from above and below.

For the sake of simplicity, in this article we have refrained from discussing semigroup generated by higher order Schrödinger operator -(-Δ)^m-V. The careful reader will have noticed that most of our results easily carry over to this more general setting, but we avoid going into details. The main ingredient for this extension would be the following.



Let V ∈ L^∞ and m ∈ℕ and consider H:=(-Δ)^m+V. For graphs satisfying Assumption <ref>, and a set K that is bounded away from σ(H). Then, it follows for z ∈ K and p ∈ [1,∞)

    ‖ (H-z)^-1‖_ℒ(L^p)≲‖ (H_ε-z)^-1‖_ℒ(L^2;H^2)≲_ε,K‖ (H-z)^-1‖_ℒ(L^2;H^2)≲_ε,K| (z) |^-1.

for the operator H_ε:= e^ε f_'(x)H e^-ε f_'(x) for a vertex ' ∈𝒱.

Fix a vertex ' ∈𝒱.


    H_ε:= e^ε f_'(x)H e^-ε f_'(x)m=0= H -ε f_'”(x)+ε^2 f_''(x)^2 -2ε f_''(x) ∂_x.

Thus, for m=0 we find that

    (H_ε-z) = (1 +  ( -ε f_'”(x)+ε^2 f_''(x)^2 -2ε f_''(x) ∂_x)(H-z)^-1_=𝒪_K(ε))(H-z)

for z ∈ρ(H). 
Analogous computations show that for arbitrary m ∈ℕ_0 and compact sets K ⊂ρ(H)

    (H_ε-z) = (1 + 𝒪_K,m(ε))(H-z).

This implies the inclusion K ⊂ρ(H_ε) too if ε≤ C_m,K for some constant C_m,K sufficiently small.
Let (χ_)_∈𝒱, be a smooth partition of unity with support (χ_) being concentrated uniformly around .
For such ε≤ C_m,K, it follows that

    ⟨χ_h, (H-z)^-1χ_'g ⟩ = ⟨χ_he^-ε f_', (H_ε-z)^-1e^ε f_' gχ_'⟩
       = e^-ε f_'()⟨χ_h e^-ε(f_'-f_'()), (H_ε-z)^-1e^ε f_' gχ_'⟩.

Thus, we have shown, using Sobolev's embedding H^2 ↪ L^p that 

    |⟨ hχ_, (H-z)^-1 gχ_'⟩|
       ≲ e^-ε d(,')‖ hχ_e^-ε(d(∙,')-d(,'))‖_L^q‖ (H_ε-z)^-1 e^εf_' g χ_'‖_L^p
       ≲ e^-ε d(,')‖ hχ_e^-ε(d(∙,')-d(,'))‖_L^q‖ (H_ε-z)^-1 e^ε d(∙,') g χ_'‖_H^2
       ≲ e^-ε d(,')‖ hχ_e^-ε(d(∙,')-d(,'))‖_L^q‖ (H_ε-z)^-1‖_ℒ(L^2,H^2)‖ e^ε d(∙,') g χ_'‖_L^2
       ≲ e^-ε d(,') e^2ε‖ hχ_‖_L^q‖ (H_ε-z)^-1‖_ℒ(L^2,H^2)‖ g χ_'‖_L^2.

Hence, we conclude from Young's inequality that

    |⟨ h, (H-z)^-1 g ⟩|   ≤∑_,' ∈𝒱|⟨ hχ_, (H-z)^-1 gχ_'⟩|
       ≲‖(H_ε-z)^-1‖_ℒ(L^2,H^2)∑_,' ∈𝒱 e^-ε d(,')(e^2ε‖ hχ_‖^q_q + ‖ g χ_'‖^p_p) 
       ≲_ε‖(H_ε-z)^-1‖_ℒ(L^2,H^2)( ‖ h‖^q_q + ‖ g ‖^p_p).

This concludes the proof.





§ PROPERTIES OF TRANSFER COEFFICIENTS

In this section we record various useful properties of the transfer coefficients introduced in Definition <ref>.
We start by collecting some properties of transfer coefficients 𝕋_,'.


For a metric graph 𝒢 and ,' ∈ℰ(𝒢) the following estimates on the transfer matrix hold

    ∑_∈ t^-1(i('))|𝕋_,'|≤ 3,   ∑_∈ t^-1(i('))𝕋_,'=1,  and 
       ∑_' ∈ i^-1(t()) c(') |𝕋_,'|≤ 3c(),   ∑_' ∈ i^-1(t()) c(')𝕋_,' =c().

Moreover, let P be a path of length m starting at an edge  then 

    ∑_P ∈𝒫_, '(m)|𝕋_P |≤ 3^m

and also 

    c()^-1∑_' ∈ℰ(𝒢)∑_P ∈𝒫_, '(m) c(') |𝕋_P |≤ 3^m .



The first set of bounds (<ref>) follow right from the definition of transfer coefficients. 
To see (<ref>), we use that for a path P of length m connecting edges  and ' we have by induction using (<ref>), assuming it to holds for paths of length m 

    ∑_P ∈𝒫_, '(m+1)|𝕋_P |= ∑_”∈ t^-1(i(')) ∑_P ∈𝒫_, ”(m)|𝕋_P ||𝕋_”,'|≤ 3^m+1.

The estimate (<ref>) follows from a similar inductive argument. Assuming it to hold for paths of length m, we find 

    c()^-1∑_' ∈ℰ(𝒢)∑_P ∈𝒫_, '(m+1) c(') |𝕋_P |   ≤ c()^-1∑_' ∈ℰ(𝒢)∑_”∈ t^-1(i())∑_P ∈𝒫_, ”(m)|𝕋_P | c(')|𝕋_”,'|
       ≤ c()^-1∑_”∈ℰ(𝒢)∑_P ∈𝒫_, ”(m)|𝕋_P |∑_' ∈ i^-1(t(”)) c(')|𝕋_”,'|
       ≤ 3 ∑_”∈ℰ(𝒢)∑_P ∈𝒫_, ”(m) c(”) |𝕋_P |≤ 3^m+1

where we used (<ref>) in the second-to-last row and the induction hypothesis in the last one.
This concludes the proof.






































§ CONVOLUTION PROPERTIES



The identity K_f *_𝒢K_g = K_f*g
holds for all f,g∈ℒ^1, where *_𝒢 is the convolution introduced in (<ref>).


This property follows by showing that 

    ∫_𝒢 K_f((ξ,),y)K_g(y,(ξ”,”))  dy = K_g*f((ξ,),(ξ”,”)).

Using (<ref>) it suffices to consider in the above integral the products of four different types of terms 

    K_h^(1a)((ξ,),(ξ','))   :=∑_P ∈𝒫_,'(m)𝕋_P h(ξ'+| P |-ξ) 
    
    K_h^(1b)(((ξ,),(ξ','))   := ∑_P ∈𝒫_,-'(m)𝕋_P h(|' |-ξ'+| P |-ξ) 
    
    K_h^(2a)(((ξ,),(ξ','))   :=∑_P ∈𝒫_-,'(m)𝕋_P h(ξ'+| P |-(||-ξ)) and 
    
    K_h^(2b)((ξ,),(ξ','))   :=∑_P ∈𝒫_-,-'(m)𝕋_P h(|' |-ξ'+| P |-(||-ξ)).

It then follows that the individual terms of K_g*f((ξ,),(ξ”,”)), i.e.,

    K_f*g^(1a) is obtained from integrands K_f^(1a)K_g^(1a) and  K_f^(1b)K_g^(1b)
       K_f*g^(1b) is obtained from integrands K_f^(1a)K_g^(1b) and  K_f^(1b)K_g^(2a)
       K_f*g^(2a) is obtained from integrands K_f^(2a)K_g^(1a) and  K_f^(2b)K_g^(2a)
       K_f*g^(2b) is obtained from integrands K_f^(2b)K_g^(2b) and  K_f^(2b)K_g^(1b),

whereas integrals involving terms

    K_f^(2b)K_g^(1b)+K_f^(2a)K_g^(2b),    K_f^(1a)K_g^(2b)+K_f^(1b)K_g^(1a) and 
       K_f^(1a)K_g^(2a)+ K_f^(1b)K_g^(1a),    K_f^(2a)K_g^(2a)+ K_f^(2b)K_g^(1a)

vanish by symmetry.




§ A GAGLIARDO–NIRENBERG INEQUALITY ON METRIC GRAPHS

Reasoning as in <cit.>, it is straightforward that the Gagliardo-Nirenberg inequality holds on the infinite graph 𝒢. We give the proof here for the sake of completeness.

Let 𝒢 be a connected locally finite metric graph with edge lengths and conductivities uniformly bounded from below and from above. Let u∈ L^2(𝒢)∩ C(𝒢) such that u^(m)∈ L^2(𝒢), then the Gagliardo-Nirenberg inequality 

    ||u||_L^∞≤ C||u^(m)||_L^2^1/2m||u||_L^2^2m-1/2m

holds on 𝒢.


By density we can consider functions u∈ C_c^∞(𝒢). Fix a vertex _0∈𝒱 and consider the subgraph G of 𝒢 induced by the vertices in a neighborhood of _0, more precisely: all vertices belonging to a ball B_r(_0) and all vertices that are adjacent to them (even if they are not in the ball), where r is chosen in such a way that supp u⊂ B_r(_0). We denote the first set of vertices by V_in and the latter one by V_out, clearly u()=0 for ∈ V_out. Now, it is possible to identify all vertices in V_out with one sole vertex of this set; in such a way we obtain a new metric graph G having multiple edges between V_out and some vertices in V_in and some loops at V_out that we can discard since u≡0 there.
 Now, for every finite graph G it is possible to construct a new graph G out of it by doubling each edge of G, such that every vertex in the resulting graph G has even degree. If  is the new edge obtained doubling  we extend u on G by u()=u(). Since G is connected and every vertex has even degree, it admits an Euler tour, i.e., a closed walk that traverses each edge exactly once. 
Therefore, G admits an Euler tour starting and terminating in V_out, and the function u:G→ℝ can be regarded as a function on the interval (0, l), where the interval (0, l) is obtained by concatenating the edges in their order of appearance during the Euler tour. Therefore, u∈ L^2 (0, l)∩ C_0(0,l) and hence we can apply the classic Gagliardo-Nirenberg inequality for functions in L^2(ℝ) with m-th derivative in L^2(ℝ) to u, c.f. <cit.>. Then,

    ||u||_L^∞(𝒢)≤||u||_L^∞(ℝ)   ≤ c||u^(m)||_L^2(ℝ)^1/2m||u||_L^2(ℝ)^2m-1/2m
       ≤ 2^|G| c||u^(m)||_L^2(G)^1/2m||u||_L^2(G)^2m-1/2m
       ≤c̃||u^(m)||_L^2(𝒢)^1/2m||u||_L^2(𝒢)^2m-1/2m

since every edge of G occurs twice in the Euler tour and conductivities are uniformly bounded from above and below.









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