Abstract
Abstract. We introduce a class of partial differential equations on metric graphs associated with mixed evolution: on some edges we consider diffusion processes, on other ones transport phenomena. This yields a system of equations with possibly nonlocal couplings at the boundary. We provide sufficient conditions for these to be governed by a contractive semigroup on a Hilbert space naturally associated with the system. We show that our setting is also adequate to discuss specific systems of diffusion equations with boundary delays.
Introduction
In the literature, usually considered problems concern networks whose ongoing dynamical processes are homogeneous: on each link, the evolution is modeled as a transport, a diffusion, a wave, a beam, etc. However, many physical models consist of coexisting, interacting processes of different type. On different edges, a different kind of dynamics may take place; or else, one may introduce fictitious, auxiliary edges in the model in order to describe certain phenomena (like delays) in a more efficient way. Accordingly, our aim is to discuss a Cauchy problem for a system of partial differential equations of mixed type: a former part of the system will satisfy heat equations, whereas a latter part will satisfies transport equations. We assume the interactions of the different subsystems to take place only at the boundary. In this way we can translate our system into a vector-valued abstract Cauchy problem with suitable, non-standard coupled boundary conditions. The topic of partial differential equations on networks has become very popular in the last fifteen years, mostly due to its connections with quantum chaos which has motivated the introduction of the keyword 'quantum graphs'. Partial differential equations on metric graphs that are associated with observables are usually studied in the framework of the theory of quantum graphs. This looks natural, considering the underlying physical motivations. So far, the large majority of investigations have been devoted to self-adjoint Laplacians and other second-order elliptic operators. We mention the comprehensive surveys [5, 18] and the article [17] . On the other hand, also some ten years ago some pioneering investigations on systems of transport equations on metric graphs have been commenced in [16] . The study of the first derivative on metric graphs inside the mathematical physics community seems to be less common and dwells on the interpretation of i d dx as the momentum operator. We are only aware of the pioneering study [6] and of the later articles [11, 15] , where some of the results of [8, 16] have been re-discovered and complemented by thorough spectral investigations.
Introduction
The aim of this paper is to connect these two theories, which have so far been studied separately. The easiest case of the coupling of one diffusion and one transport equation (both possibly vector-valued) has been considered by Gastaldi and Quarteroni in [12] . This work was presented by its authors as a first step towards the coupling of a Euler and a Navier-Stokes equation. A related viscosity analysis has been performed in [2] .
Introduction
In [12] , well posedness of the problem has been studied by means of viscosity methods under certain (relatively strict) coercivity assumptions. We are going to weaken the coercivity assumptions and study a general system consisting of finitely many coupled intervals of either type. We are going to do so by considering a large class of coupled boundary conditions for operator matrices including both second and first derivatives.
Introduction
While the proposed dynamics actually seems to reflect the observed phenomena, it is very difficult to make an educated guess when it comes to propose natural transmission conditions. It seems that the search for correct transmission conditions, which is easy in the case of purely diffusive [19] or pure transport-like systems [16] , is much less trivial in the mixed case. As Gastaldi and Quarteroni put it in [12, section 1] , 'When coupling Euler and Navier-Stokes equations, the proper interface conditions are not obvious, in advance. A possible way of deducing them is to see the coupled problem as a limit of two coupled Navier-Stokes problems with vanishing viscous terms (in either one of the regions)'. But even this limiting process is quite delicate, as the analysis in [2, 12] shows.
Introduction
Our main result, theorem 4.1, states that systems of the kind discussed above are well posed under a large class of transmission conditions. This may hopefully suggest suitable conditions in specific settings. Furthermore, we complement a spectral analysis of the considered system of typical features of semigroups for heat and transport equations. In particular, contractivity of a semigroup guarantees that the norm of solutions to the initial value problem are not larger than the norm of the initial data.
Introduction
Unlike in [12] , a possible motivation for the introduction of our setting arises from some biomathematical considerations. More precisely, it is known that electric signal coming from a neuron undergoes a certain synaptic delay before reaching another neuron, and cannot turn back: this suggests to model this process by a system of diffusion (in the dendrites or axons) and transport (in the synapse) equations. This is explained in some detail in section 2.
Introduction
In section 3, we introduce the specific class of boundary conditions we are going to investigate. In section 4, we discuss, in dependence of the boundary conditions, some sufficient conditions for well posedness of the problem in the L 2 -setting, collecting all proofs in section 5. In section 6, we derive a secular equation for our system and an explicit representation of the resolvent.
Introduction
It turns out that our setting can be adapted to discuss certain classes of coupled systems of diffusion equations with boundary delays. We describe in Subsection 4.1 how to apply our abstract results to the motivating problem from Section 2.
Introduction
Most of the present paper has been written with a main focus on evolution equations of parabolic type. Even when the evolution equation is not governed by an analytic semigroup, we still think of our problem as a sort of degenerate diffusion equation. For this reason, in this paper we investigate some properties that are typical for diffusive problems -and less so for quantum mechanical ones. In order to keep this article as self-contained as possible, we recall in the appendix all the definition and abstract notions we are going to need. We refer to [10] for more details and proofs.
Delayed diffusion equations via mixed PDEs on graphs
Let us discuss a concrete example, which involves the problems outlined in the introduction and suggests a possible interplay with delayed diffusion equations. It is a classical idea, thoroughly developed e.g. in [3] , that delays can be mathematically modeled by means of (auxiliary) transport phenomena. While this theory is classical whenever the delay is 'distributed' -i.e. it acts on each point of the domain of diffusion -it is less standard and technically more involved whenever the delays only concern the boundary values. The situation we want to describe is the following. We consider a ramified structure with an ongoing diffusion process. Incoming particles are absorbed in some nodes and stored there for some time (which depends solely on the features of the node itself), before triggering a flow in the incident edges. This phenomenon can be described by attaching to each such node a fictitious loop, which the particles have to cross before reaching the adjacent edges. A comparable approach has been presented in the recent article [4] .
Delayed diffusion equations via mixed PDEs on graphs
As a tentative motivation of the investigation of this class of problems, we briefly discuss a (much simplified) model of a dendrodendritic chemical synapse, referring the reader to any introductory textbook on neuronal modeling (e.g., [20, chapters 2 and 9] and [7, chapter 5] ) for basic notions and some miscellaneous models, including delayed ones. Also for the sake of simplicity, we drop the absorption term and therefore replace the cable equation usually discussed in theoretical neuroscience by a simpler diffusion equation. (This is in fact a bounded perturbation; hence it can be neglected when it comes to discussing well posedness). We also remark that our model essentially applies to the case of less exotic axodendritic synapses, if we linearize the active transport phenomena typical of the latter. Of course, although the situation considered here is motivated by a toy neuronal model, it can also be regarded as a simple example of a general diffusion equation with boundary delays.
Delayed diffusion equations via mixed PDEs on graphs
Consider two dendrites (modeled by two intervals e 1 , e 2 ) that are incident in the synapse v, which is terminal endpoint of e 1 and initial endpoint of e 2 . The synaptic input coming from e 1 undergoes a delay of τ del = 1 before reaching e 2 and cannot turn back. For the sake of simplicity, we also impose sealed end conditions on the first dendrite e 1 as well as on the second endpoint of e 2 :
Delayed diffusion equations via mixed PDEs on graphs
Of course, longer chains of neurons might be modeled in a similar way. The synaptic input is an action potential that lets neurotransmitters be released by synaptic vesicles. Experimental observations seem to suggest that no obvious (linear) algebraic relation exists between the pre-and post-synaptic potential in the dendrites -i.e. between the boundary values of the unknowns u 1 and u 2 in the diffusion equations. We propose to discuss this system by a network diffusion problem with boundary delay
Delayed diffusion equations via mixed PDEs on graphs
The first boundary condition, of Robin-type, can be interpreted by saying that part of the ions reaching the presynaptic nerve terminal is reflected into the entrance dendrite. The second condition is actually the relevant one. It reflects the fact that the conduction speed of the signal in the dendrites is approximatively constant, at least within the same cortical area. In other words, even if the postsynaptic potential will in general have a different amplitude from the presynaptic one, their speed of propagation will be the same. Finally, it is easy to convince ourselves that the above problem is undetermined if the last initial condition on the delay term is not imposed.
Delayed diffusion equations via mixed PDEs on graphs
In order to implement the delay feature into the node conditions, we introduce an edge e del and an unknown u del to model the transport of neurotransmitters in the synaptic cleft between the pre-and the postsynaptic neurons, hence effectively introducing a delay phenomenon:
Delayed diffusion equations via mixed PDEs on graphs
Our aim is now to replace the fourth, i.e. the delayed equation in (BD) by two node conditions in the endpoints of e del . More precisely, we impose that
Delayed diffusion equations via mixed PDEs on graphs
as well as
Delayed diffusion equations via mixed PDEs on graphs
In other words, the flow of postsynaptic potential and the flow of presynaptic potential agree, even if its transmission undergoes a certain delay (which we have normalized).
Delayed diffusion equations via mixed PDEs on graphs
We are hence led to consider an (undelayed) initial boundary value problem
Delayed diffusion equations via mixed PDEs on graphs
i.e. we have got rid of the boundary delay by passing to the a larger state space. Observe that the above model is intrinsically non-symmetric in the sense that potential can only flow from dendrite e 1 to e 2 , but not vice versa. This is a typical feature of chemical synapses, as opposed to electric ones. One can check that the problems (BD) and (BD ) are equivalent. It should be observed that transportbased synaptic transmission models are not very common in the literature. Due to their numerical and analytic complexity they are actually often replaced by convective-diffusive (or even purely diffusive) models. A convincing pleading of a transport approach can be found in [21, sections 2 and 6]. In the following, a general framework is discussed that allows us to deal not only with this motivating example.
Boundary conditions for a mixed operator on a metric graph
We use the standard construction for the singular manifold on which the dynamic of the system is going to take place: we consider finitely many intervals of finite length that are connected to realize a simple finite metric graph G with node set V and edge set E. For the purposes of this paper, it is important to consider a partition of E in two disjoint subsets E d and E t , which are going to represent the edges on which diffusion and transport phenomena are going to take place, respectively. Hence, we denote by e d1 , . . . , e dD and e t1 , . . . , e tT the edges of the metric graph. To each edge e di or e tj , we associate a length a di or a tj , respectively, which in turn determines an orientation of each edge from 0 to the other endpoint. More specifically, this orientation leads to the representation
Boundary conditions for a mixed operator on a metric graph
and in this case we write e(0) = v, e(a) = w. (Here and in the following, a denotes the generic lengththat is, either a di or a tj , depending on the context.) The structure of the network is given by the |V|×|E|-outgoing and ingoing incidence matrices I + := (ι For the sake of notational simplicity, if e = − −− → (v, w), we denote the value of a function u e : [0, a] → C on e at 0 and a by u e (v) and u e (w), respectively. With an abuse of notation, we also set u e (v) = 0 whenever e / ∈ Γ(v). By assumption, E is a finite measure space. The space L 2 (E) of square integrable functions defined on the intervals associated with the edges in E becomes a Hilbert space with respect to the natural scalar product
Boundary conditions for a mixed operator on a metric graph
, we denote the L 2 (E)-functions with support on the edges belonging to E d and E t , respectively. Note that any element u ∈ L 2 (E) admits the representation
Boundary conditions for a mixed operator on a metric graph
Finally, by H m (E d ) and H m (E t ) we denote the Sobolev spaces of square integrable functions that are m-times weakly differentiable on each edge with weak derivatives in L 2 (E d ) and L 2 (E t ), respectively. We stress that we are not imposing boundary conditions of any type on the functions in
Boundary conditions for a mixed operator on a metric graph
e (w) = 0 for any edge e =
Boundary conditions for a mixed operator on a metric graph
, we formally consider the diagonal operator matrix
Boundary conditions for a mixed operator on a metric graph
We are going to present a class of m-dissipative realizations of A whose domains contain the minimal one
Boundary conditions for a mixed operator on a metric graph
: the fact that these are the extreme relevant cases follows from the observation that in both cases the transport and the diffusion part are clearly edge-wise decoupled.
Boundary conditions for a mixed operator on a metric graph
is a Hilbert space with respect to the natural inner product
Boundary conditions for a mixed operator on a metric graph
, is a Hilbert space with respect to the inner product
Boundary conditions for a mixed operator on a metric graph
Their embedding into L 2 (E) is dense and of pth Schatten class for all p > 1. Furthermore, there exists C > 0 such that the Gagliardo-Nirenberg-type estimate
Boundary conditions for a mixed operator on a metric graph
holds.
Boundary conditions for a mixed operator on a metric graph
Proof. Throughout this paper, we are assuming our graph to be compact, hence the embeddings of the Hilbert spaces
Boundary conditions for a mixed operator on a metric graph
are of pth Schatten class for all p > 1, cf [13] .
Boundary conditions for a mixed operator on a metric graph
A constant C > 0 can always be found so that (3.4) holds, as this inequality can be reduced to the case of intervals. If namely u ∈ H 1 (0, 1); then
Boundary conditions for a mixed operator on a metric graph
Indeed if u ∈ H 1 (0, 1) is such that u(0) = 0, then we can write
Boundary conditions for a mixed operator on a metric graph
and by Cauchy-Schwarz's inequality, we get
Boundary conditions for a mixed operator on a metric graph
For a general u ∈ H 1 (0, a), it suffices to apply the previous estimate to u(x) = (1 − We now impose conditions that force the numerical range to lie in a left half-plain of the complex plain. An elementary integration by parts yields (3.6)
Boundary conditions for a mixed operator on a metric graph
, where we have introduced a notation based on the (not sign definite) sesquilinear forms
Boundary conditions for a mixed operator on a metric graph
While the equation (3.6) is not particularly appealing, the real part of the associated form is
Boundary conditions for a mixed operator on a metric graph
Re Au, u = −
Boundary conditions for a mixed operator on a metric graph
For the sake of notational simplicity we introduce the 2|E d | + |E t | dimensional ("boundary") Hilbert space
Boundary conditions for a mixed operator on a metric graph
, the two vectors u, u ∈ H, where
Boundary conditions for a mixed operator on a metric graph
are given with respect to the decomposition of H explained in (3.8) . In particular, with this specific representation for the boundary values, equation (3.7) can be compactly re-written as (3.10) Re Au, u = −
Boundary conditions for a mixed operator on a metric graph
These computations motivates us to introduce a class of boundary conditions of the form
Boundary conditions for a mixed operator on a metric graph
where P is an orthogonal projector acting in H, P ⊥ := Id −P denotes the complementary orthogonal projector and the matrix L is an operator in the subspace KerP ⊂ H (whose extension by 0 to the whole of H we still denote by L). The boundary conditions can be equivalently written as P ⊥ u + Lu = 0 and P u = 0.
Boundary conditions for a mixed operator on a metric graph
We finally define the operator A P,L which is studied in this work as the operator A equipped with the boundary conditions defined in (3.11), i.e.,
Boundary conditions for a mixed operator on a metric graph
Hence, inserting (3.11) into (3.10) yields
Main results
We are finally in the position to state the main result of this article. The proof will be postponed to Section 5.
Main results
Theorem 4.1. Let P be an orthogonal projector acting on H and L a linear operator on KerP such that −L satisfies the condition
Main results
for some ω ≥ 0. Then the operator A P,L defined in (3.12) is quasi-m-dissipative, and in fact m-dissipative whenever −L is dissipative. Accordingly, the operator A P,L is the infinitesimal generator of a quasi-
Main results
. If in addition P and L have only real entries, then the semigroup generated by A P,L is real.
Main results
Clearly, (4.1) is satisfied whenever −L is dissipative -but not only, as the examples in subsections 4.1 and 4.2 show.
Main results
The above theorem yields in particular that the initial value problem
Main results
is well posed whenever −L is dissipative or even satisfies (4.1), and the solution is given by
Main results
Furthermore, the question whether the solution u(·, t) is a real-valued function for all t > 0 if the initial data u 0 is real-valued can be answered in terms of the boundary conditions imposed at the vertices.
Main results
Remark 4.2. We call the boundary conditions (3.11) type-decoupling if
Main results
• P is an orthogonal projection of H onto {0}, H, C 2|E d | × {0}, or {0} × C |Et| , and additionally • L is a block-diagonal matrix with respect to the decomposition of H into C 2|E d | and C |Et| .
Main results
In other words, the boundary conditions are type-decoupling if actually no interaction between the boundary values in L 2 (E d ) and L 2 (E t ) takes place: this is of course the most trivial case, because the dynamics of the system can be effectively reduced to that of two distinct, non-interacting systems -a diffusive one and a transport one.
Main results
One sees that the semigroup generated by A P,L is not irreducible if the boundary conditions (3.11) are type-decoupling. In particular, for type decoupling boundary conditions theorem 4.1 reproduces the classical results known for the diffusion equation and the transport equation, which have so far been studied separately, as recalled in the introduction.
Main results
We conclude by stating a conjecture on the time-dependent behavior of solutions to the initial value problem considered here.
Main results
If a function f is supported in e ∈ E t , then the semigroup generated by A P,L will shift its profile until the support of e tA P,L f hits an endpoint of e. Because e tA P,L f has in this lapse of time the same profile of f , the semigroup cannot be immediately smoothing. In particular, it cannot be analytic -in fact, not even immediately differentiable. On the other hand, it seems reasonable to imagine that the semigroup smoothens the profile of a function as soon as it reaches an edge in E d . We conjecture that if E d = ∅, then the semigroup is differentiable for all t > T , i.e. (T, ∞) t → T (t)x ∈ X is differentiable for all x ∈ X, where T is the length of the longest path inside E t (to compute taking into account the possibly coupled boundary conditions). Since
Main results
is not sign definite, −L is not dissipative. However, one has
Main results
Decomposing the matrix as above is critic, as this allows to find norms
Main results
and subsequently to estimate
Main results
This shows that (4.1) is satisfied and a direct application of theorem 4.1 yields the following.
Proposition 4.3. The initial-boundary value problem (BD ) is governed by a strongly continuous semigroup
. Furthermore, this semigroup is real.
Proposition 4.3. The initial-boundary value problem (BD ) is governed by a strongly continuous semigroup
Remark 4.4. Observe that the above result does not really depend on our choice to consider a delay interval e del of unit length: we may in fact replace (0, 1) by an interval of arbitrary finite length. The same holds for the lengths of the diffusion edges. Moreover, −L is not dissipative; hence theorem 4.1 cannot be applied to deduce contractivity of the semigroup that governs the problem.
Another example.
Observe that dissipativity of the matrix −L is sufficient but not necessary for A P,L to be m-dissipative. To give a concrete example of an m-dissipative operator A P,L where −L is not dissipative, we consider the graph consisting of one transport and one diffusion edge with certain lengths a d1 and a t1 , e d1 e t1
Another example.
and boundary conditions as in ( 
Inserting this into (3.7) delivers
Re A P,Lα u, u = −
Inserting this into (3.7) delivers
Note that the form defined by
Inserting this into (3.7) delivers
is dissipative for α > 0 small enough. Since for
Inserting this into (3.7) delivers
for a constant C > 0, and hence Re A P,Lα u, u ≤ 0, for
Inserting this into (3.7) delivers
Since L α satisfies (4.1) for some ω > 0 for α > 0 small enough one has by theorem 4.1 that the operator A P,Lα is m-dissipative for α > 0 small.
4.3.
A different model. Instead of considering the transport-diffusion-type Cauchy problem associated with dissipative operators, one might think of a Schrödinger-type Cauchy problem involving self-adjoint operators with mixed dynamics. To this aim, consider the symmetric operator
It has equal deficiency indices (2|E
and its adjoint S = (S 0 ) * is formally the same operator with domain
It has equal deficiency indices (2|E
Hence there exists self-adjoint extensions and these can be parametrized in terms of boundary conditions. One defines the appropriately modified vectors of boundary values byū
It has equal deficiency indices (2|E
to obtain the well known Hermite symplectic form on the space of boundary values
It has equal deficiency indices (2|E
It is known that there is a one-to-one correspondence between the self-adjoint extensions of symmetric operators and the maximal isotropic subspaces with respect to this Hermite symplectic form, see e.g. [14] . A unique parametrization of such a subspace is given in terms of a projection P and a Hermitian operator L acting in KerP . Therefore, any self-adjoint realization S P,L of S 0 is a restriction of S to a domain of the type
It has equal deficiency indices (2|E
The spectral theory for the operators S P,L can be developed on the lines of the spectral theory for the operators A P,L elaborated in section 6. In particular, it follows like in the proof of lemma 6.1 below that the resolvents of the self-adjoint operators S P,L are compact, and hence their spectrum is purely discrete. Furthermore, one can obtain an explicit expression for the resolvents like that in the forthcoming proposition 6.6.
It has equal deficiency indices (2|E
The self-adjoint operators S P,L can be interpreted as Hamiltonians consisting of a standard Laplacian and a less usual moment-type observable. As already mentioned, moment operators on graphs have been recently studied in [11] . By Stone's theorem, for self-adjoint S P,L the abstract Cauchy problem
It has equal deficiency indices (2|E
is governed by a unitary group.
Proofs of the main results
Before proving theorem 4.1, we need two preparatory lemmata.
Proofs of the main results
Lemma 5.1. Let ω ≥ 0 be such that (4.1) is satisfied. Then A P,L is quasi-dissipative: more precisely,
Proofs of the main results
where C > 0 is a constant, depending only on the total length of the graph, such that
Proofs of the main results
The estimate (5.2) follows from (3.4).
Proofs of the main results
Proof. Take u ∈ D(A P,L ). Then it follows from (3.10) that
Proofs of the main results
If instead L only satisfies (4.1) for ω > 0, then it follows from (3.10) that
Proofs of the main results
for all ε > 0. In particular, for ε = 2 ωC we obtain
Proofs of the main results
This concludes the proof.
Proofs of the main results
It is straightforward to verify that the adjoint of A 0 in the Hilbert space L 2 (E) is the operator
Proofs of the main results
As A P,L ⊂ A is an extension of B * = A 0 , the operator A * P,L is a restriction of B. Therefore, it can be described in terms of boundary conditions imposed on B. We introduce the notation
Proofs of the main results
where
Proofs of the main results
(The change of sign in the last component is due to the change of the direction on the transport edges).
Proofs of the main results
Lemma 5.2. The adjoint operator of A P,L is given by
Proofs of the main results
Proof. By definition, the adjoint of A P,L is the operator given by
Proofs of the main results
To begin with, observe that following the computations in (3.6) the operators A and B -without boundary conditions -satisfy
Proofs of the main results
Recall that A * P,L is a restriction of the operator B. Hence v ∈ D(A *
Proofs of the main results
is Ker(L + P, P ⊥ ) and hence the boundary term vanishes for a fixed
Proofs of the main results
taking into account that
Proofs of the main results
Since the orthogonal complement of the space Ker(−P ⊥ , L+P ) is exactly Ker(L * +P,
Proofs of the main results
This completes the proof.
Proofs of the main results
Proof of theorem 4.1. It is known ([9, corollary II.3.17]) that a sufficient condition for a densely defined operator to have m-dissipative closure is that both the operator and its adjoint are dissipative. By lemma 5.1, the operator A P,L is quasi-dissipative for any L satisfying (4.1) for some ω ≥ 0 and dissipative whenever −L is dissipative. Like in lemma 5.1 one proves that conversely the operator A * P,L is quasi-dissipative for any L satisfying (4.1) for some ω ≥ 0 and dissipative whenever −L is dissipative using
Proofs of the main results
To conclude the proof, it suffices to check that A P,L is actually closed. Because both the first and the second derivative without boundary conditions are closed operators, in our case it suffices to check that the boundary conditions are respected in the limit. This follows from the fact that u → u and u → u are bounded operators from the Hilbert space H 2 (E d ) ⊕ H 1 (E t ) to H. In order to prove reality of the semigroup it is sufficient to show that the resolvent maps real function to real functions. The resolvent is calculated explicitly in proposition 6.6 in the forthcoming section 6 as an integral operator. Observe that its kernel r(·, ·, iκ), κ > 0 has real coefficients whenever the operator P and L has real entries. Therefore, the resolvents (A P,L − κ 2 ) −1 map real-valued functions to real-valued functions for any κ 2 > 0. Applying the generalized inverse Laplace transform described in (7.1), one concludes that the semigroup generated by A P,L is real whenever P and L are real.
Spectral theory
As we have mentioned in the appendix, cf (7.1), knowing the resolvent of A P,L one can describe the strongly continuous semigroup generated by it by means of the inverse Laplace transform, for A P,L quasim-dissipative.
Spectral theory
In the following the spectrum of the operator A P,L is analyzed and an explicit formula for the resolvent is derived. Taking into account lemma 3.1, we promptly obtain the following.
Spectral theory
Lemma 6.1. For all orthogonal projectors P on H and all linear operators L acting on KerP satisfying (4.1) the operators A P,L have resolvent of pth Schatten class for all p > 1 (and, in particular, of HilbertSchmidt class). In particular, the operators A P,L have only pure point spectrum.
Spectral theory
6.1. Non-zero eigenvalues. In order to determine the pure point spectrum of A P,L , a natural Ansatz for finding eigenfunctions is to take k ∈ C \ {0} and to consider
Spectral theory
x ∈ e tj , j = 1, . . . , T.
Spectral theory
where
Spectral theory
..,T = γ tj (k) are the sought after coefficients. The matrices
Spectral theory
acting in H are given with respect to the decomposition
Spectral theory
Here, the notation {e and {e
Spectral theory
is used, where δ jl is the Kronecker delta. Accordingly, the following holds.
Spectral theory
Proposition 6.2. For all orthogonal projectors P in H and all linear operators L acting in KerP , the number −k 2 ∈ C \ {0} is an eigenvalue of A P,L if and only if the matrix
Spectral theory
has non trivial null space. The geometric multiplicity of −k 2 equals the dimension of KerZ P,L (k).
Spectral theory
Hence, the secular equation is det Z P,L (k) = 0. In general, it seems to be difficult to give precise statements on the distribution of the eigenvalues. Then the operator A P,L is m-dissipative and the secular equation becomes
Spectral theory
= 0.
Spectral theory
In particular, the spectrum of A P,L contains a sequence of real eigenvalues going to −∞.
Eigenvalue zero.
For the eigenvalue zero, one uses for the eigenfunctions the Ansatz
Eigenvalue zero.
x ∈ e tj , j = 1, . . . , T.
Eigenvalue zero.
The boundary conditions (P + L)φ(·, 0) + P ⊥ φ(·, 0) = 0 are encoded in
Eigenvalue zero.
where
Eigenvalue zero.
Hence we obtain a characterization for the eigenvalue zero.
Eigenvalue zero.
Proposition 6.4. For all orthogonal projectors P in H and all linear operators L acting on KerP , the operator A P,L has eigenvalue zero if and only if
Eigenvalue zero.
In particular, the invertibility of the operator A P,L for L satisfying (4.1) is independent of the lengths of the transport edges. For the situation considered in subsection 4.1, for example, one has that the operator A P,L is not invertible for any edge lengths.
Eigenvalue zero.
Example 6.5. Consider as in example 6.3 the graph consisting of one diffusion edge of length a d1 and one transport edge of length a t1 . Let
Eigenvalue zero.
where
Eigenvalue zero.
and therefore A P,L C is invertible for Re C ≥ 0.
Eigenvalue zero.
6.3. The resolvent operator. In the following, an explicit formula for the resolvent is given in terms of the boundary conditions and the edge lengths. First we define the shorthand notation
Eigenvalue zero.
Proposition 6.6. For all orthogonal projectors P in H, all linear operators L acting on KerP and for all
Eigenvalue zero.
with integral kernel
Eigenvalue zero.
Here we have denoted
Eigenvalue zero.
and furthermore
Eigenvalue zero.
Finally,
Eigenvalue zero.
where the entries are diagonal matrices whose entries are functions with arguments from the corresponding edges and
Eigenvalue zero.
Proof. It is sufficient to prove that r(x, y, k) defines the Green's function of the operator (A P,L + k 2 ). Consider the unperturbed operator
Eigenvalue zero.
The equation (A + k 2 )R 0 (k)u = u is satisfied on the diffusion edges as 
Eigenvalue zero.
and for all This gives in the matrix notation for u ∈ L 2 (E) and v ∈ H
Eigenvalue zero.
Therefore,
Eigenvalue zero.
and hence for all u ∈ L 2 (E),
Eigenvalue zero.
This proves that r(·, ·, k) is the Green's function for the mixed problem.
Appendix: A reminder of semigroup theory
In this final section, we are going to recollect some results from the general theory of strongly continuous semigroups, in order to make the technique of this article more transparent to the reader less familiar with it. In particular, we are interested in the case of generators that are neither skew-adjoint nor self-adjoint. We refer to [1, 9, 10] for a comprehensive introduction and overview to modern semigroup theory.
Appendix: A reminder of semigroup theory
Let in the following H be a complex Hilbert space with scalar product ·, · and induced norm · . It is well known that all strongly continuous semigroups (T (t)) t≥0 of bounded linear operators on a Hilbert space are exponentially bounded, i.e. there exist constants M ≥ 1 and ω ∈ R such that T (t) ≤ M e ωt for all t ≥ 0.
Appendix: A reminder of semigroup theory
Furthermore, a semigroup is called
Appendix: A reminder of semigroup theory
• ω-quasi-contractive, for some ω ∈ R, if T (t) ≤ e ωt for all t ≥ 0.
Appendix: A reminder of semigroup theory
• quasi-contractive if it is ω-quasi-contractive for some ω ∈ R, and finally • contractive if it is ω-quasi-contractive for ω = 0. For some ω ∈ R, a closed and densely defined operator A is called ω-quasi-dissipative if Re Au, u ≤ ω u 2 for all u ∈ D(A).
Appendix: A reminder of semigroup theory
For a given ω ∈ R, the operator A is called ω-quasi-m-dissipative if it is ω-quasi-dissipative and additionally λ − A is surjective for some λ > ω. An operator A is called quasi-m-dissipative (resp., quasi-dissipative) if it is ω-quasi-m-dissipative (resp., ω-quasi-dissipative) for some ω ∈ R.
Appendix: A reminder of semigroup theory
By the Lumer-Phillips Theorem, a semigroup is ω-quasi-contractive if and only if its (necessarily closed and densely defined) infinitesimal generator A is ω-quasi-m-dissipative, cf [1, theorem 3.4.5]. While the above range condition can be sometimes hard to check directly, it is known that an operator is ω-quasi-mdissipative if in particular both the operator and its adjoint are ω-quasi-dissipative.
Appendix: A reminder of semigroup theory
In addition to their central role in the Lumer-Phillips Theorem, quasi-m-dissipative operators are important since it is possible to represent the semigroup generated by them by means of a suitable generalization of the inverse Laplace transform. More precisely, if A is the ω-quasi-m-dissipative infinitesimal generator of the semigroup T (t), then the formula Let us now consider the case where H is a complex-valued L 2 (X)-space for some σ-finite measure space (X, µ). Then a bounded linear operator T on H is called
Appendix: A reminder of semigroup theory
• real if T f (x) ∈ R for µ-a.e. x ∈ X, whenever f (x) ∈ R for µ-a.e. x ∈ X;
Appendix: A reminder of semigroup theory
• irreducible if it does not leave invariant any non-trivial ideal of L 2 (X) -i.e. it does not leave invariant the subspace L 2 (X) of L 2 (X) for any measurable subsetX of X withX = ∅ andX = X.
Appendix: A reminder of semigroup theory
An operator semigroup (T (t)) t≥0 on H = L 2 (X) is called real (resp. irreducible) if each operator T (t) is real (resp. irreducible).

{'paper_id': '119315063', 'title': 'Quantum graphs with mixed dynamics: the transport/diffusion case', 's2_url': 'https://api.semanticscholar.org/CorpusID:119315063', 'source': 'S2ORC'}