On Open Scattering Channels for a Branched Covering of the Euclidean Plane
Rainer Hempel Olaf Post
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Open Scattering Channels for a Two-Sheeted Covering]
On Open Scattering Channels for a Branched Covering of the Euclidean
  Plane.


Institute for Computational Mathematics, TU Braunschweig,
  Universitätsplatz 2, 38106 Braunschweig, Germany
r.hempel@tu-bs.de


Mathematik, Fachbereich 4, Universität Trier, 54286 Trier, Germany
olaf.post@uni-trier.de




[2010]35P, 35Q, 81U


   We study the interaction of two scattering channels for a simple
   geometric model consisting in a double covering of the plane with 
   two branch points, equipped with the Euclidean metric. 
   
  We show that the scattering channels are open in the sense of <cit.> 
 and that this property is stable under suitable perturbations of the metric.




[
    Olaf Post
    December 30, 2023  File: 
=============================









§ INTRODUCTION





Let M denote a branched covering of the plane, obtained by glueing
two copies of ^2 along a straight-line cut between the points = (-1,0) and = (+1,0), where the northern edge of the upper
copy of ^2 is joined to the southern edge of the lower copy, and
vice-versa (see 1). The branch points  do not
belong to M.

The manifold M is a real version of the complex Riemann surface
associated with the function √(z^2 - 1).  With the Euclidean
metric  of ^2, we obtain a smooth, connected Riemannian
manifold = (M,) with curvature zero; note, however, that
 is not complete. In the second part of the paper we will
consider Riemmannian metrics g on M which are close to  in
a suitable sense so that the perturbational results of <cit.>
can be applied.

We let H denote the Laplacian of , a self-adjoint operator
acting in the Hilbert space =.  For a metric g on
M, different from the Euclidean metric, we denote the associated
Laplacian by H_g.  It is the aim of this paper to study some
asymptotic properties of the unitary groups (^- t H ; t ∈) and (^- t H_g ; t ∈).  In particular, we are
interested in the question whether there is transmission from the
lower to the upper sheet and vice versa. As noted by Percy Deift 
(private communication), this amounts to the question


  “When I shout on the lower plane, will I be heard on the
    upper plane?”



For the comparison dynamics (with two scattering channels) we take the
free Laplacian on two copies of ^2 which we may imagine to lie one
atop of the other. In other words, we consider the Hilbert space
_0 = ^2⊕^2 and we let H_0 denote the
direct sum of two copies of the self-adjoint Laplacian in
^2,

    H_0 = H_0,⊕ H_0,,

where the indices  and  mean “lower” and “upper,”
respectively. H_0 is (purely) absolutely continuous. With a natural (unitary) identification J _0
→ the wave operators

    W_±(H,H_0,J) = e^ tH J e^- tH_0,

exist, are complete, and isometric, as will be seen in
wave-ops-euclid. Since also H is absolutely continuous 

the wave operators W_±(H,H_0,J) are in fact unitary. 
Writing J = J_⊕ J_, the
channel wave operators W_±(H, H_0,, J_) and
W_±(H, H_0,, J_) are given by

    W_±(H, H_0,k, J_k)
         = e^ tH J_k e^- tH_0,k, 
              k ∈{, }.

Note that f ∈ W_+(H, H_0,, J_) means that there
exists h ∈_ such that

    ^- t H f - J_^- tH_0, h → 0, 
           t → +∞;

in particular, ^- tHf is asymptotically in the upper
sheet, as t → +∞.  This leads to the question whether states
which come in on the lower sheet will also go out on the lower sheet,
or whether there are states which change sheets as t goes from
-∞ to +∞. We construct, indeed, states that move from
the lower to the upper sheet, up to a small error.  It follows that
there is non-zero transmission between the upper and the lower sheets
of , or, in the terminology of <cit.>, that the upper and
the lower channels are open. By symmetry there is also transmission
from the upper to the lower sheet; since it is more or less trivial
that there is transmission within the two sheets we find that all
scattering channels are open one to another. This is stated as
1.9.

We next ask whether the scattering channels remain open when the
Euclidean metric  on M is replaced with a more general
metric g on M which is close to  at infinity in the sense
of <cit.>.  The corresponding assumptions concern, in
particular, the harmonic radius <cit.>)
and the injectivity radius of (M,g), and the difference of the
Riemannian metrics  and g in a suitable distance function.
Here we profit in several ways from the fact that the geometry of
 is so simple.  We require that the metrics g and  be
quasi-isometric in the usual sense (cf. 2.2), and we
assume a global bound on the curvature of (M,g). Under additional
assumptions on g, expressed in terms of the distance
d̃_1(,g) in eqn. (<ref>), 2.3 states that
the wave operators

    W_±(H_g, H_, I_g) :=^ tH_g I_g ^- tH_
 
exist and are complete, where H_ = H is the Laplacian of
(M,), H_g is the Laplacian of (M,g), and I_g is the
natural identification between M, and M,g; as 
was mentioned earlier, H_ is purely absolutely continuous. 

In 2.3, smallness of the perturbation is only required at
infinity. In contrast, for the question of openness of the scattering
channels the deviation of g from  has to satisfy a global,
quantitative smallness condition. Then 2.4 establishes the
strong convergence of the scattering operators

    S(H_g_, H_0, I_g_ J) 
        :=  (W_+(H_g_, H_0, I_g_J))^* 
                    ∘ W_-(H_g_, H_0, I_g_J)
 
to S(H, H_0, J) for a sequence of metrics g_ on M tending to
 as ↓ 0.  In 2.5 we then obtain the
openness of all scattering channels for small .

The paper is organized as follows. In wave-ops-euclid  we
introduce most of our notation and we discuss some basic spectral
properties of the manifold = (M,), deferring the details 
and proofs to Appendix A. We then turn to
scattering for the pair (H, H_0) where we establish existence and
completeness of the wave operators. The technically difficult part of 
wave-ops-euclid concerns the construction of a wave packet that
comes in from infinity on the lower sheet and moves out to infinity on
the upper sheet. Here we use ideas from Enß' theory of scattering
and stationary phase estimates to construct states that pass between
the branch points  at time t = 0 at high speed, and which are
essentially localized to a double cone.

In pert-metric we consider metrics g on M that are close
(or, at least, close at infinity) to the Euclidean metric . In
essence, we only have to write down what the basic definitions and
results of <cit.> mean in the present context. We then find
simple conditions for the existence and completeness of the wave
operators (<ref>) as well as for a non-trivial
interaction between the scattering channels for (M,g).


The main results of pert-metric are illustrated in
examples by a simple class of metrics on M, namely metrics g
= g_f that come from the graph of smooth functions f on M.  It
turns out that it is fairly easy to indicate conditions on f so that
the metric g_f satisfies the requirements of 2.3.  We finally
discuss branched coverings with more than two sheets and corresponding
generalizations of the present results.

 The paper comes with three appendices; the first two of them
 are mainly included for the convenience of the reader.
 spectral-properties is
 devoted to self-adjoint extensions, compactness and spectral
 properties of the Laplacian with metrics  and g.
 As for the absolute continuity of H_ and H_g,
 we mainly refer to some work of Donnelly <cit.>
 and Kumura <cit.>.
 
In stationary-phase we recall a basic estimate from stationary
phase theory to establish an estimate on the localization error for
the Schrödinger evolution. More precisely, for suitably chosen
initial data u_0 in the Schwartz space ^2 we
multiply u(t) := ^ t Δ u_0 by a cut-off function χ
and obtain estimates for ∇χ·∇ u(t) and (Δχ) u(t) in the -norm.

lower-bound-inj-rad is devoted to lower bounds for the
injectivity radius of (M,g) where the metric g on ^2 or on M
is close to the Euclidean metric. Starting from a comparison result of
Müller and Salomonsen <cit.> we obtain
“local” versions by means of cut-offs and extension theorems,
proceeding from ^2 via ^2 ∖{(0,0)} to M.

We conclude the introduction with a few remarks concerning the literature.
The paper <cit.> and the literature quoted there give
a partial overview of Riemannian scattering on manifolds with ends. 
Recent progress in this direction can be found in Güneysu and Thalmaier
 <cit.>. The specific case of manifolds with
branch points has been studied in recent years under various aspects and
our results have some overlap with the work of Hillairet and others;
cf. <cit.>  and <cit.>.
There is a connection between the analysis of the Aharonov-Bohm effect
in Quantum Mechanics and branched coverings of Euclidean space;
cf. <cit.>. Scattering for magnetic Schrödinger operators
with two magnetic point charges has been studied in a number of papers; as
an example, we mention Ito and Tamura <cit.> which has
some connection with our investigations. 




 §.§ Acknowledgements



The authors thank Luc Hillairet (Univ. d'Orléans) for an 
interesting discussion and comments. Rainer Hempel would like 
to express his gratitude to Brian Davies
(King's College, London), Percy Deift (Courant Institute, New York),
Ira Herbst (Univ. of Virginia, Charlottesville), Barry Simon
(Caltech, Pasadena), and Larry Thomas (Univ. of Virginia) for
valuable discussions and suggestions concerning the matter of the
present paper.






§ WAVE OPERATORS FOR THE EUCLIDEAN METRIC

  



Let us begin with some notation. As far as general notation
for self-adjoint operators T in a Hilbert space  is concerned we
mostly follow <cit.> and <cit.>.  In particular,
we let _(T) denote the absolutely continuous subspace of 
associated with T, and P_(T) the orthogonal projection onto
_(T).  For the general formal setup of multi-channel
scattering we refer to Section 4 of <cit.> and the literature
quoted there.  Since the model studied in the present paper is so
simple, we develop most notions in multi-channel scattering directly
as we go along.

 
Let M be defined as in the Introduction. We then denote the points
 of M by ((x,y), ) or ((x,y),) where (x,y) ∈^2 and “” means “lower,” “” means “upper”.
This works for all points of M with the exception of the points with
-1 < x < 1 and y = 0; note that these exceptional points form a
set of measure zero. With  denoting the metric tensor = (δ_ij) we obtain the Riemannian manifold :=
(M,).  For the remainder of this section we will be cavalier
about the distinction between M and =(M,) and we will
mostly write M.  For two points _1, _2 ∈ M the
(geodesic) distance is then given by

    (_1,_2) 
      := infγγ(0) = _1, γ(1) = _2

where γ [0,1] → M is a rectifiable curve and γ denotes the length of γ.  It will be useful to extend
the definition of distance to the branch points  and .  The
infimum in (<ref>) is attained either for a straight line
segment connecting _1 and _2 or for (the union of) two straight line
segments that meet at one of the branch points.  E.g., if _1 =
((0,y), ), _2 = ((0,-y),) with y > 0, then
(_1,_2) = (_1, ) + (,_2) = 2 √(1
  + y^2) (see 2 left).


For a point _0 ∈ M, we denote the (geodesic) disc of radius 
r > 0 and center (x_0, y_0) by B_r(_0), i.e., 

    B_r(_0) = ∈ M(, _0) < r;

such discs may or may not contain points in both sheets (see 2
right), and they may even contain pairs of points (p, p') with the
same (x,y)-coordinates and p in the lower, p' in the upper
sheet.  A disk B_r(p_0) will be “single-valued” if and only if r
≤min{(p_0, q_+), (p_0,q_- }. In the extreme case of
p_0 ∈{q_+, q_-} and 0< r ≤ 2 the disk B_r(p_0) will just
be a double covering of the punctured disk (x,y) ∈^20 <
  x^2 + y^2 < r^2.  

The Riemannian manifold  is not (geodesically) complete.

In order to define the Laplacian H of , we consider the Hilbert
space := with scalar product denoted by ··, and the Sobolev space , given as the completion of
M with respect to the norm [1] · defined by

    [1] ψ
        := ∫_M ψ(x) + ∇ψ(x) x, 
            ψ∈ M.
  
Then H is defined as the unique self-adjoint operator satisfying
(H) ⊂ and

    Hu v 
      = ∫_M ∇ u ·∇ v x, 
           u ∈(H),     v ∈.

It is easy to see (cf. spectral-properties) that 
coincides with the Sobolev space = [2] 1,
consisting of all functions in  that have first order
distributional derivatives in . Hence the Laplacian on
M has only one self-adjoint extension with form domain
contained in .  However, the Laplacian is not
essentially self-adjoint on M.  Basic spectral properties of
H are also discussed in spectral-properties; in particular,
H is purely absolutely continuous with H = [ac] H =
[0,∞).

We next consider the Rellich compactness property.  For the proof we
refer to A.2 in spectral-properties.


  
  For R > 0, let χ_R denote the characteristic function of M_R
  = B_R() ∪ B_R() ⊂ M (see 2).  Then the mapping
  ∋ u ↦χ_R u ∈ is compact.



We now turn to scattering theory and introduce the comparison dynamics
for the scattering channels associated with the two sheets (and two
infinities) of M.

Let M_0 := ^2 ⊎^2 = ^2 ×{, } denote the
disjoint union of two copies of the Euclidean plane ^2, and write
M_0, = ^2 ×{}, M_0, = ^2 ×{}.  We then let _0 = M_0, = ^2⊕^2 Moreover, we let H_0 denote the Laplacian on
M_0. To fix the notation, let A_0 denote the (unique) self-adjoint
extension of -Δ on ^2. We may then write H_0 =
H_0,⊕ H_0, where H_0, and H_0, act
as A_0 in M_0,, and in M_0,,
respectively.

We denote the straight line segment in ^2 connecting the points
 as Γ,

    Γ := [-1,1] ×{0}⊂^2,
 
a set of measure zero.  There is a natural embedding ι
(^2 ∖Γ) ×{, }→ M, ι =
(ι_, ι_), where ι_ maps the point 
((x,y), ) ∈ M_0,∖Γ to ((x,y),) ∈ M, 
 and similarly for ι_. 
 The embedding ι induces a unitary mapping J
_0 → where J = J_⊕ J_ in an obvious
manner (and with a slight abuse of notation).  J_ maps functions
f ∈M_0,, to the same function on the lower sheet of M
and extends them by zero to all of M, and similarly for J_.
We then have:



  
  The wave operators
  
    W_±(H,H_0,J) =^ tH J ^- tH_0
 
  exist and are unitary. 




 
  As is often the case in two Hilbert space
    scattering <cit.>, there is a certain
    arbitrariness in the choice of the mapping J. 
    By local compactness, the same wave
    operators and the same results would be obtained with J replaced
    by (1 - χ_R)J, for some R > 0, or by (1 - ϕ)J
    with an  arbitrary ϕ∈^2. 
 





  We decouple both H and H_0 by Dirichlet boundary conditions
  along two circles defined as follows. Let _2 := (x,y) ∈^2x^2 + y^2 = 4, _2' := _2 ×{,}⊂ M_0, and _2” := ι( _2') ⊂
  M. Introducing Dirichlet boundary conditions on _2' and on
  _2” decomposes H_0 into a direct sum of four operators
  while H is decomposed into a direct sum of three operators. More
  precisely, we introduce the following three “building blocks:” in
  the plane ^2, we have the Dirichlet Laplacian h_ on
  the disc of radius 2 and the Dirichlet Laplacian h_ on the
  exterior of this disc.  Furthermore, defining
  
    
    
    M_0,   :=  (x,y) ∈^2x^2 + y^2 > 4×{, },
    
          M_    := ι(M_0,)), 
             M_ := M ∖ M _

  
  we denote by H_ the Dirichlet Laplacian of
  M_. Note that M_ is a branched covering with
  two sheets of the punctured disc (x,y) ∈^2x^2 + y^2
    < 4∖{q_+, q_-}.  We then write
  
    
    
    H_0,    := (h_ , ) ⊕ (h_, ) 
            ⊕  (h_ , ) ⊕ (h_, ),
    
          H_    := H_⊕ (h_ , ) ⊕ (h_ , );

  
  note that h_ is purely absolutely continuous while
  h_ and H_ (by 1.2) have compact
  resolvent.

  It is well-known (<cit.>) that the wave operators
  
    W_±(H_0,,H_0) =^ tH_0,^- tH_0

  exist, are complete, and isometric with initial subspace
  _(H_0) = _0 and final subspace _(H_0,) =
  M_0,,.
 Similarly, it can be shown by standard methods (cf. <cit.>), that the wave operators
  
    W_±(H, H_) =^ tH^- tH_ P_(H_)

  exist, are complete, and partially isometric with initial subspace
  _(H_) = M_, = _(H_0,) =
  M_0,, and final subspace _(H) = M,.
 
  Finally, the wave operators
  
    W_±(H_, H_0,,J)
        =^ t H_ J ^- t H_0, P_(H_0,)

  simply act as the identity on M_,, and as the zero
  operator on M_,.  Therefore, they exist and are
  complete.
 It is now clear that the wave operators W_±(H,H_0,J) exist and are unitary. 



With J = J_⊕ J_ and H_0, as defined above, we
furthermore see that the channel wave operators

    W_±(H, H_0,, J_)
      = ^ tH J_^- tH_0,

(and, analogously, W_±(H, H_0,, J_) exist and are isometric  with 

    W_±(H, H_0,, J_) ⊕ W_±(H, H_0,, J_)
      =   W_±(H, H_0, J)
      = _(H);
 

recall that f ∈ W_+(H, H_0,, J_) means that there
exists g ∈M_0,, such that

    ^- t H f - J_^- t H_0, g→ 0, 
           t →∞;
 
in particular, ^- t H f is asymptotically on the lower sheet
for t →∞.  Eqn. (<ref>) establishes two orthogonal
decompositions of _(H) = M,, one for the plus-sign
and another one for the minus-sign. We will see later on
(cf. 1.7.2nd.part) that these two decompositions are in fact
different.



  
  Let us note that H_0 = H_0,⊕ H_0, provides a
  reference operator for H in the sense of <cit.>
  with two channels.  Strictly speaking, branch points like  are
  not directly included in the framework used
  in <cit.>. However, this technical difficulty is easy to
  resolve: we might just take each of the sets B_1/2() as an
  end, albeit an end which does not participate in the scattering
  process since the Dirichlet Laplacian of B_1/2() has compact
  resolvent by 1.2. The possibility of allowing such “dead
  ends” is described in Remark 4.4 of <cit.>. We thus have
  (formally) a manifold with 4 ends, with two ends given by a copy of
  ^2 ∖ B_2(0) and another two ends given by
  B_1/2().




It is a major goal in scattering theory to obtain information on the
scattering operator

    S = S(H,H_0,J) 
      := (W_+(H,H_0,J))^*∘ W_-(H,H_0,J) _0 →_0,

a unitary operator, and the closely related scattering matrix
(S_ij)_i,j ∈{, }, with

    S_ij 
      := (W_+(H, H_0,i, J_i))^* ∘ W_-(H, H_0,j, J_j)
                               M_0,j,→M_0,i,,

for i, j ∈{, }. We will show that the four components
of (S_ij) are non-zero which yields the openness of all scattering
channels. 

The following lemma establishes the existence of a state w_0 for
which ^- tHw_0 is asymptotically in the lower sheet for t
→ -∞ and in the upper sheet for t → +∞, up to small
errors. Recall that A_0 denotes the self-adjoint extension of the Laplacian
on ^2. We then have:



  
  For > 0 given, there exist w_0 ∈M,∩[∞](M), v_0 ∈^2, and t_0 ≥ 0
  such that the following estimates hold:
  
    ^- tH w_0 - J_^- t A_0 v_0 < w_0, 
             t ≥ t_0,

  and 
  
    ^- tH w_0 - J_^- t A_0 v_0 < w_0, 
              t ≤ -t_0.

 





In the proof of 1.7 we basically construct a state v_0 ∈^2 which passes at high speed between the points 
under the evolution determined by ^- tA_0 (up to small
errors) and whose spreading can be controlled by stationary phase
estimates, for |t| large.  Note that we have complete control of the
unitary group (^- t A_0 ;  t ∈), acting in
^2, while we know much less about (^- tH ;  t
∈), acting in M,.  By a simple lifting, v_0 is
transformed into a function w_0 on M.  Here we wish to gain
information on the evolution of ^- t H w_0 from the
properties of ^- t A_0 v_0 using the fact that both operators
act locally as the Laplacian.

Recall that H_0, and H_0, denote the self-adjoint
Laplacian in M_0,, and in M_0,,,
respectively.  We let  denote the Fourier transform on the
Schwartz spaces ^d for d ∈.  It is well known
that  acts bijectively on ^d and extends to a
unitary map ^d→^d.




Our construction starts with a function u_0 ∈^2 of
the form u_0 = u_0(x,y), given as the product of two functions
ψ_1= ψ_1(x) and ψ_2 = ψ_2(y) enjoying certain
properties, which we describe now.

Let ∈ (0,1) be given and let ' := /5.  We first
pick a function ϕ_1 ∈ of norm 1 and we let ψ_1 :=
^-1ϕ_1 ∈ where we assume that

    χ_(-1/4,1/4)ψ_1 > 1 -'.
   
We let a = a_ > 0 be such that ϕ_1 ⊂ (-a,a).
Next, let ϕ_2 ∈ C_c^∞ (0,1), of norm 1 again, and let
ψ_2 := ^-1 [ ϕ_2(.-s)] ∈, where s > 0
will be chosen later.  Let

    u_0 = u_0(x,y) = ψ_1(x) ψ_2(y), 
           (x,y) ∈^2.

Then u_0 ∈^2⊂(A_0) and u(t) := ^-
  tA_0 u_0 is a classical solution of the initial value problem for
the Schrödinger equation in ^2, i.e.,

    u̇(t)
      = - A_0 u(t) for t ∈ (0,∞),
           u(0) = u_0.


We write

    Q_s,t := 
       
        (-st, st) × (st, ∞),      t > 0, 
    
        (st, -st) × (-∞, st),     t < 0,
 
for s > 0, and we let χ_s,t denote the characteristic
function of Q_s,t.  a.2 implies that for any m ∈
there exists a constant c_m ≥ 0 such that

    (1 - χ_s,t) ^- t A_0 u_0≤c_m (1 + st)^1 - 2m, 
           s ≥ 2a,    t > 0,

so that for s ≥ 2a and t large, t ≥ t_0 say,  

    ( 1 - χ_s,t) ^- t A_0 u_0≤'.



Now let

    Ω := (x,y) ∈^2 x < 1/2 +  y,

let χ_Ω denote the characteristic function of Ω,
and, finally,

    χ := j_1/4 * χ_Ω,

where (j_δ)_δ > 0 is the kernel of the usual Friedrichs
mollifier on ^2; in particular, 0 ≤ j_δ∈^2
with support in the closed disc of radius δ, and ∫ j_δ
= 1.  Also let  denote the support of χ and X the
characteristic function of , i.e, X = χ_. Note that
χ is independent of t.



We next consider v_0 := χ u_0 and observe that the (smooth)
function v := χ u is a solution of the
inhomogeneous initial value problem

    v̇(t) = -  A_0 v(t) +  f(t), 
           v(0) = v_0,

with f = f(t) = f(x,y;t) given by

    f = - 2∇χ·∇ u -  u Δχ.

We also have v_0 - u_0 < ' and v_0 > 1 - '.
Stationary phase estimates (cf. a.3 in the appendix) imply that
there exists s_0 ≥ 0 such that

    ∫_-∞^∞f(τ)τ < ', 
           s ≥ s_0.
 
The solution v = v(t) of eqn. (<ref>) can be written as

    v(t) = ^- t A_0 v_0 + ∫_0^t ^ (t-τ) A_0 f(τ) τ.
 
Notice that there is no reason to expect that for t  0 the
individual terms ^- t A_0 v_0 and ∫_0^t ^ (t-τ)
  A_0 f(τ) τ on the right-hand side of (<ref>)
should vanish outside of ; it is only the sum of the two terms
which has support contained in .  It is immediate from
eqn. (<ref>), u_0 - v_0 < ', and u_0 =
1 that

    (1 - χ_s,t) ^- t A_0 v_0≤ 2 ',  
          χ_s,t^- t A_0 v_0≥ 1 - 2'.


We have now gathered all the information we need on ^-itA_0 v_0
and are ready for the proof of 1.7.


 
  
  
  * 
    In order to make the transition from ^2 to M we define a map
    j → M which assigns to (x,y) ∈ the point
    ((x,y),) ∈ M for y < 0, and the point ((x,y),) ∈
    M for y > 0.  The points in  with y = 0 are mapped to
    the line segment where the lower and the upper sheets of M are
    connected as we move in the direction of increasing values of
    y. Let ' := j(). For functions η→,
    we obtain a lifting J η' → defined by
    
    ( Jη)(j(x,y)) := η(x,y), 
               (x,y) ∈.
 
    We may extend J η by zero to all of M.  Obviously, we
    have w(t) :=  J v(t) ∈(H) for t > 0 and H (w(t)) =
     J A_0(v(t)).  Hence w is a classical solution in  of the initial value problem
    
    ẇ(t) = - H w(t) +  Jf(t),
               w(0) =  Jv_0,
 
    so that
    
    w(t)
          = ^- tH Jv_0 
               + ∫_0^t ^ (t-τ)H J f(τ) τ.

    We conclude from eqns. (<ref>) and (<ref>) that
    
    0     = w(t) -  Jv(t)
        = e^- tH Jv_0 
               + ∫_0^t ^ (t-τ)H Jf(τ) τ 
               -  JX ^- t A_0 v_0 
               -  J X ∫_0^t ^(t-τ) A_0 f(τ) τ,

    whence
    
    [] ^- tH Jv_0 -  J X ^-it A_0 v_0≤ 2 ∫_0^t f(s) s < 2'.

    
    We finally define w_0 :=  Jv_0 and note that
    w_0 > 1 - '.

  
  * 

    We now prove eqn. (<ref>).  Combining (<ref>)
    and (<ref>) we see that
    
    ^- tH w_0 - J_u ^-it A_0 v_0   ≤^- tH w_0 -  J X ^-it A_0 v_0 
            + ( J X - J_u) ^-it A_0 v_0
       ≤ 2 ' + (1 - χ_s,t) ^-it A_0 v_0
       ≤ 4 ' < 4 '/1 - 'w_0
            < w_0,

    since ( J X - J_u) χ_s,t^-it A_0 v_0 = 0 for t >
    0, 0 <  < 1, and ' = /5.

    The proof of (<ref>) is similar and omitted. 



It is now easy to prove the main result of this section. 


  
  The entries of the scattering matrix (S_ij)_i,j ∈{,
    }, as defined in Eqn. (<ref>), are all non-zero
  operators.





  
  
  * 
    We first show that the operator S_ is non-zero.  Let
    0 <  < 1/4 and let v_0 and w_0 be as in
    1.7. Without loss of generality we may assume, in addition,
    that w_0 = v_0 = 1. Then
    
    S_ v_0v_0 
          = W_-(H,H_0,, J_) v_0W_+(H,H_0,, J_) v_0
 
    where, by 1.7,
    
    W_-(H,H_0,, J_) v_0 - w_0 < ,     W_+(H,H_0,, J_) v_0 - w_0 < .

    It now follows that |S_ v_0v_0 - 1| ≤ 3
     < 3/4. This shows that S_ is non-zero; but
    then, by symmetry, we also have S_ 0.


  
  * 
    In order to show that S_ (and, analogously,
    S_) is non-zero, it is enough to construct wave
    packets which come in on the lower sheet (limit t → -∞)
    and which go out on the lower sheet as well (limit t →
    +∞), up to a small error. It is easy to modify v_0 and
    w_0 as in 1.7 to achieve this goal; cf. also 1.10
    below.  E.g., we may replace the function ψ_1 in the proof
    of 1.7 with ψ_1(· - k) with k > 1 so that
    the associated wave packet is located away from the slit at time
    t = 0.  We then translate Ω, χ, and  in the
    x-direction accordingly.  The maps j and J can be simply
    defined as an embedding of  into M_0,. We leave the
    details to the reader. 
 




  
  In fact, what we obtain here is a particularly strong version of
  openness of the channels in the sense that the norm of the wave
  packet going out on one sheet is close to the norm of the incoming
  state on the other sheet, for suitably chosen states.  For example,
  for any > 0 there are states where the norm of the outgoing
  wave packet on the upper sheet is greater than (1 - ) times
  the norm of what is coming in on the lower sheet, etc. One might say
  then that the channels are strongly open.





   
  In dealing with S_ we might as well exchange the
  variables x and y and translate in the y-direction to avoid
  the slit. In the end, all one needs is a rigid motion of  which
  avoids the slit and one gets the impression that “most” initial
  states will belong to the range of S_ or S_
  while only a tiny fraction of initial states communicates between
  the two sheets under the evolution ^- tH. Thus, if one
  wishes to be heard on the upper plane as a member of the lower plane
  one should shout in the right direction (and also rather at a high
  pitch).




Here we give some indications on coverings of the
Euclidean plane with three or more sheets. In the case of three
sheets and two branch points the southern rim of the cut in the sheets numbered
I, II, and III is identified with the northern rim of the sheets numbered
II, III, and I. Then the situation is basically the same as with two sheets
and all channels are open. In the case of four sheets and two
branch points the identification of the rims proceeds as above.
Here we can show that neighboring sheets are open to one another while our
method fails to decide whether the sheets I and III are open
one to another; the same holds for the sheets II and IV. We suspect that
the transmission is very weak (or zero) in the latter cases. 

For three and more sheets there are of course also other possibilities
to connect the sheets along cuts. For three sheets we might look at
two different cuts (and thus four branch points) with sheets I and II
connected along the first cut and sheets II and III connected along
the second cut.  If the two cuts are not aligned we may still
construct wave packets that move from sheet I up to sheet III, up to
small errors. If the two cuts are aligned (i.e., both lie on the real
axis and have positive distance) our method fails. In this last case
we would expect that there is only very weak (or no) transmission from
sheet I to sheet III.

Also note that we are dealing with two (or more) branch points because
a manifold with two sheets and a single branch point—like the
Riemann surface of √(z)—constitutes just one scattering channel
in our setup.  In this case there is no simple comparison with the
free Laplacian on the Euclidean plane.






   
  The singularities at the branch points are only a side issue in our
  investigations. For most of our results, it wouldn't make much of a
  difference if we would “punch out” two small holes around the
  branch points and consider the Laplacian with Dirichlet boundary
  conditions on the (smooth) boundaries of these balls.  However, the
  radius of these balls would introduce a parameter which is not well
  motivated and one would have to investigate questions of convergence
  etc. as this radius goes to zero.




For the record, we complement the estimates of 1.7 with some
further basic properties of w_0.


   
  Let P_±, and P_±, denote the projections onto
  the ranges of the wave operators W_±(H, H_0,, J_) and
  W_±(H, H_0,, J_), respectively.  For > 0 let
  w_0 be as in 1.7.  We then have:
  
    P_+, w_0 > (1 - ) w_0,     P_+, w_0 < w_0 .

   and
   
    P_-, w_0 > (1 - ) w_0,     P_-, w_0 < w_0.

  

 
 We only show (<ref>); the proof of (<ref>) is 
 analogous and ommitted. By the Projection Theorem,
    we have
    
    P_±,  w_0    = 
          supw_0ψψ∈ W_±(H, H_0,, J_),   ψ = 1
        =
          supw_0W_±(H, H_0,, J_) ϕϕ∈_,  ϕ = 1,

    since W_±(H, H_0,, J_)ϕ = ϕ for
    all ϕ∈_.  In the RHS of eqn. (<ref>) we
    have
    
    [] w_0W_±(H, H_0,, J_) ϕ
          = lim_t →±∞[]^- t H w_0J_^- t H_0,ϕ.

    In order to obtain a lower bound on P_+,  w_0 we
    choose ϕ := v_0 in Eqn. (<ref>) and use 1.7 to
    find
    
    [] ^- t H w_0 J_^- t H_0, v_0
                                  - w_0^2   < .

    For an upper bound on P_+,ℓ w_0 we use
  J_ J_ℓ = 0, combined with 1.7, to see that  
    P_+, ℓ w_0 <. 



Of course, one could as well work with the usual formula for the
projection onto the range of a partial isometry. In our case this
formula reads

    P_±,  
       =  W_±(H, H_0,, J_) ∘(W_±(H, H_0,, J_))^*.




Let us first show that the adjoints (W_±(H, H_0,, J_))^* of the 
wave operators W_±(H, H_0,, J_) are given by strong limits,

    (W_±(H, H_0,, J_))^* 
      =^ t H_0, J_^* ^- t H,

with P_(H) = I.  Since the wave operators W_±(H, H_0, J)
exist and are complete (and because J satisfies the requirements
of <cit.>, it follows that the wave
operators

    W_±(H_0, H, J^*)
      = ^ tH_0 J^* ^- tH

exist. Here H_0 = H_0,⊕ H_0, and J^* = (P_,
P_) and we see that

    ^ tH_0 J^* ^- tH
      = ( ^ t H_0, P_^- t H, 
        ^ t H_0, P_^- t H).

The ranges of ^ t H_0, P_^- t H and
^ t H_0, P_^- t H being orthogonal, it is
clear that the strong limit of the left hand side of (<ref>)
can only exist if the strong limits of both terms on the right hand
side exist (as t →±∞).


For w_0 as in 1.7 we now compute

    P_±, w_0w_0
       = lim_t →±∞^ t H_0, J_^* ^- t H w_0
       = lim_t →±∞P_^- tH w_0

and the desired result follows by 1.7.





§ PERTURBATIONS OF THE METRIC

 



We first recall some notions and definitions in Differential Geometry
as used in <cit.>. Given a (smooth) Riemannian metric g =
(g_ij) on the -manifold M, we denote by =(M,g) the
Riemannian manifold and we let B_δ(p)=B_δ,(p) denote
the geodesic open ball centered at p ∈ M with radius δ >
0. For simplicity, we only consider smooth metrics g on M;
cf., however, the discussion in <cit.> on the non-smooth
case. Our assumptions on g will mainly involve the (sectional or
Gauß) curvature of g and the injectivity radius. The
homogenized injectivity radius ι_(p) at p ∈ M is
defined as in <cit.> or <cit.>
by
 
    ι_(p) := sup_δ > 0min{δ, 
             inf_(y)y ∈ B_δ,(p)}

where _(y) denotes the usual injectivity radius at the point
y.  The number ι_(p) is the largest number δ > 0 for
which the injectivity radius at any y ∈ B_δ(p) is not smaller than δ.


The following definition (cf. <cit.>) is of basic importance for our investigations:


  
  For a continuous positive function r_0  M → (0,1] we
  denote by _r_0(M) the set of smooth metrics g
  on M that satisfy the lower bounds
  
    ι_(p) ≥ r_0(p), andinf^-_(y)y ∈ B_r_0(p),(p)≥  - 1/r_0(p)^2,
   
  for all p ∈ M, where =(M,g).


Since we are in two dimensions, the Ricci curvature equals the Gauß
curvature (times the metric tensor g). The second
condition in eqn. (<ref>) is a lower bound for the homogenized
Ricci curvature.

Notice that the Euclidean metric = (δ_ij) on M 
 belongs to _r_0(M) if and only if r_0 satisfies the condition

    r_0(p) 
      ≤1/2min{(p,), (p,) }.
  

We denote by  the usual space of (equivalence classes of)
-integrable functions on the Riemannian manifold
=(M,g) with respect to the Riemannian measure _g.  The
following definition is standard.


  
  We say that the Riemannian metrics g_1, g_2 are
  quasi-isometric if there exists a constant η > 0 such
  that
  
    η g_1(p)(ξ,ξ) ≤ g_2(p)(ξ,ξ) ≤η^-1 g_1(p)(ξ,ξ),

  for all ξ∈ TM and p ∈ M. 


In our case TM can be identified with ^2.  The Hilbert spaces
_1 and _2 coincide if _i = (M, g_i) with
g_1 quasi-isometric to g_2. In this case we let I denote the
natural identification operator mapping a function f ∈_1 to the same function f in _2.

We now take a closer look at the property that the metrics g and
 are quasi-isometric.  Let A(p) be the endomorphism on TM
given by g(p)(ξ,ξ)=(A(p)ξ,ξ) for all ξ∈ T_pM
and p ∈ M and let α_k(p), k = 1,2, denote the eigenvalues
of A(p).  If (g_ij(p)) denotes the matrix representation of g
on T_pM in the standard coordinates, then the α_k(p) are
also the eigenvalues of (g_ij(p)).  Thus g and  are
quasi-isometric if and only if there is a number η > 0 such that
η≤α_k(p) ≤η^-1, for k = 1, 2 and for all p
∈ M.

We are now ready to define the basic distance function d_1: Let
α_1(p), α_2(p) denote the eigenvalues of A(p).  We
then define as in <cit.>

    d (,g)(p)
      := max_k α_k(p)^1/2 - α_k(p)^-1/2,
  

    d_∞(,g) := sup_p ∈ M d(,g)(p)
      and d_1(,g) := ∫_M   d(,g)(p) r_0(p)^-4 p.

We call d_1(,g) the weighted
  [1]-quasi-distance of g and ; we have dropped
the symmetrizing factor 1 + ρ_,g(p) of d_1
appearing in <cit.> (which has no influence on our
estimates because it is a bounded function).

Let us assume now that g is quasi-isometric to the Euclidean metric
 (this is equivalent with d_∞(g,)<∞),
and denote by =(M,g) the corresponding Riemannian manifold.  Then
there is a (unique) self-adjoint Laplacian H_g, acting in the
Hilbert space , with quadratic form domain given by the
Sobolev space , and defined by

    H_g u v
       = ∫_M [g]∇ u∇ v_g
       = ∑_ij∫_M g^ij∂_i u ∂_j v√( g) p,

for any u ∈(H_g) ⊂ and v ∈,
where (g^ij) is the inverse of (g_ij).  In the Euclidean case
(g =) the operator H_ agrees with the operator H
defined in wave-ops-euclid; recall that H_ is purely
a.c. From Theorem 3.7 of <cit.> we now obtain the following
result on the existence and completeness of the wave operators.



  
  Suppose we are given a continuous function r_0 M (0,1]
  satisfying condition (2.3) and a metric g ∈_r_0(M) which
  is quasi-isometric to the Euclidean metric  on M.
  We also assume that the difference between g and 
  satisfies the r_0-dependent weighted integral condition
  d_1(g,) < ∞ with d_1 as in (<ref>).

  Then the wave operators
  
    W_±(H_g,H_,I) =^ tH_g I ^- tH_
 
  and
  
    W_±(H_g,H_0,IJ) =^ tH_g IJ ^- tH_0
 
  exist and are complete with final subspace _(H_g). 




  Under suitable conditions on g the operator H_g will be
  absolutely continuous (cf. Donnelly <cit.>,
  Kumura <cit.>. In this
  case the wave operators in (<ref>) are even unitary.
 


  In applying the fundamental perturbation theorems in <cit.>
  we can deal with the branch points  in the way described in
  1.5, i.e., we have (formally) a manifold with four ends, with
  two ends given by M_ as in Eqn. (<ref>) and two ends
  given by B_1/2().  Again, the ends B_1/2() do not
  participate in the scattering.



Following the development in Section 5 of <cit.> we next
consider the question of continuity of the scattering matrix and the
openness of the scattering channels for small perturbations of the
Euclidean metric. As in <cit.> we define for r_0 as above and
γ,  > 0

    _r_0(M, , γ, ) 
      := g ∈_r_0(M) d_∞(g,) ≤γ,    d_1(g,) ≤,

i.e., _r_0(M, , γ, ) is the set of smooth metrics
g on M enjoying the following properties:


  * 
  The  homogenized injectivity radius and the homogenized curvature of g  
  at p ∈ M are bounded from below by r_0(p) and by 
  -1/r_0(p)^2, respectively. 

  * 
  The metric g is quasi-isometric to  with the bound d_∞(g,) ≤γ.


  * 
  The weighted [1]-quasi-distance d_1(g,) is
  not larger than .

Note that condition def.met.iii requires a quantitative
smallness of the deviation of g from the Euclidean metric in the
sense that d_1(g,) ≤ while the main assumption in
2.3 only stipulates d_1(g,) < ∞.

Then Theorem 5.1 of <cit.> yields the strong convergence of the
scattering operators as ↓ 0, and Cor. 5.3
of <cit.> establishes the openness of the scattering channels,
for small > 0. We are now going to make this precise.

Let γ > 0 be fixed. For > 0, we consider g_∈_r_0(M, , γ, ) and we let H_g_ denote
the Laplacian of (M,g_). The natural identification operator
from M, to M,g_ is written
I_g_. Then the scattering operator is given by

    S_g_
      = S(H_g_, H_0, I_g_J) 
      = (W_+(H_g_, H_0, I_g_J))^* 
                    ∘ W_-(H_g_, H_0, I_g_J),
   
with H_0 and J as in wave-ops-euclid, 1.4, and the scattering
matrix (S_ij)_i,j ∈{,} is defined by

    S_ij(H_g_, H_0, I_g_J)
      := (W_+(H_g_, H_0,i, I_g_J_i))^* 
                    ∘ W_-(H_g_, H_0,j, I_g_J_j),

for i,j ∈{, }. Then <cit.> yields the
following result:



  
  Let , H_, H_0, J as above, let S(H_,H_0,J) as in
  eqn. (<ref>) , and let γ > 0.  For
  > 0 and g_∈_r_0(M, , γ, ) we
  denote by H_g_ the Laplacian of _ = (M, g_) and
  by I_→_ the natural
  identification.

  Then there exists _0 > 0 such that the scattering operators
  S(H_g_, H_0, I_ J) converge strongly to S(H, H_0, J),
  as → 0.



As in <cit.>, we immediately obtain a stability
result for the scattering matrix
where we also use the fact, established in 1.9, that the
operators S_ik(H, H_0, J), i,k ∈{, }, are non-zero, 
i.e., all scattering channels are open.  



  
  For any γ > 0 fixed, there exists _0 > 0 such that
  S_ik(H_g_, H_0, I_ J)  0 for all metrics g_∈_r_0(M, , γ, ) and all 0< ≤_0.





§ EXAMPLES





We first illustrate 2.3 in the special case where the 
perturbed metric on M is associated with the graph of a 
function f  M → of class [2]. As usual, 
we define Φ M →^3 by Φ(p) := (p,f(p)) and  

    g = g_f = J_Φ^T· J_Φ 
      =
      [ 1 + f_x^2   f_x f_y;   f_x f_y 1 + f_y^2;           ],

 where J_Φ is the Jacobian of Φ.
The eigenvalues of g are 1 and g = 1 + f_x^2 + f_y^2. 

The curvature κ of := (M,g) is given by the well-known formula

    κ
      = f_xx· f_yy - f_xy^2/(1 + f_x^2 + f_y^2)^2 
      =  H_f/^2 g

(cf. <cit.>, <cit.>), 
where H_f is the Hessian of f. We let 

    d_0(p) := min{1, (p,), (p,) } ,      p ∈ M,

where the distances are measured in (M,). We have the following
proposition.

 
   
  Let f M be of class [2] with bounded
  first and second order derivatives and suppose that
  
    ∫_M |∇ f|^2  d_0^-4 p < ∞.

  Then the wave operators W_±(H_g, H_, J) exist and are complete. 
 


  Since g = g_f has the eigenvalues 1 and 1 + f_x^2 + f_y^2 with
  f_x, f_y bounded, the metric g is quasi-isometric to the
  Euclidean metric on M.  We now choose a suitable function r_0
  which then defines the class _r_0(M).
  Note that the choice of r_0 is not unique, and one may 
  obtain different results for different choices. 
  In view of eqn. (<ref>) the simplest choice appears to be
  r_0 := ρ  d_0 with a constant ρ∈ (0, 1/2] which we are
  going to fix now.

  Since f has bounded second order derivatives, the curvature of
  (M,g_f) is bounded in absolute value by some constant K ≥ 0
  and the second condition in Eqn. (<ref>) is satisfied
  provided ρ≤ 1/√(K).  According to b.5 there exists
  a constant c_0 > 0 such that the (homogenized) injectivity radius
  of (M,g) at p ∈ M is bounded from below by c_0 d_0(p). We
  may thus pick any ρ > 0 satisfying ρ≤min{1/2, 1/√(K), c_0}.

  In remains to show that d_1(,g) as in eqn. (2.6) is
  finite. Here we first estimate
  
    d(,g) = √(1 + f_x^2 + f_y^2) - 1/√(1 + f_x^2 + f_y^2)≤ 
        |∇ f|^2;

  therefore condition (<ref>) implies d_1(,g) <
  ∞. Since the assumptions of 2.3 are satisfied, we may
  conclude that the wave operators for the pair (H_,H_g) exist and
  are complete.





  
  
  * Condition (<ref>) is satisfied if ∇ f is
    square integrable at infinity and decays near  and  like
    
    |∇ f(p)| ≤min{(p,), (p,) }^1 + β

    for some β > 0.

  
  * It is illuminating to take a look at other choices of r_0
    where r_0 tends to zero at infinity.  The class of admissible
    functions f  M → that define the perturbed metric
    changes in the following way. On the one hand, the injectivity
    radius associated with the metric g may now go to zero at
    infinity and the (Gauß) curvature need no longer be bounded
    from below by a constant; on the other hand, it is now more
    difficult to satisfy the weighted integral
    condition (<ref>).
  



In an analogous way one can indicate simple conditions on f which
allow the application of 2.4. We consider functions f  M
→ of class [2] with first and second order
derivatives bounded by some constant C and which are such that g_f
∈_r_0(M) with r_0 as above.  Then d_∞(g_f,
) ≤ C^2, and we may now choose γ := C^2.  For >
0, the condition d_1(g_f,) ≤ is safisfied if

    ∫_M |∇ f|^2  d_0^-4 p ≤;

 in this case, we have g_f ∈_r_0(M, , γ, ) and
 the results of 2.4 and 2.5 apply.

 
  Suppose we are given a sequence (f_n) ⊂[2] M enjoying
  the following properties:
  
  
  * There is a constant C ≥ 0 such that |∂_j f_n(p)|
    ≤ C and |∂_ij f_n(p)| ≤ C for all p ∈ M and
    all n ∈.

  
  * We have
    
    ∫_M |∇ f_n|^2  d_0^-4 p  → 0,      n →∞.

  
  Let g_n denote the metric induced by f_n and let I_n the
  associated natural identification operator, as above.  
  
  Then the scattering operators S(H_g_n, H_0, I_n J) exist and
  converge strongly to S(H_, H_0, J), as n →∞.
 
















§ SELF-ADJOINTNESS AND SPECTRAL PROPERTIES.

  






In this appendix we study the Sobolev spaces ^1
and Laplace-Beltrami operators on branched coverings of the
Euclidean plane. Here we are mainly interested in self-adjointness,
compactness properties, and the question of absolute continuity of the
Laplacian.





 §.§ Double covering with a single branch point.


 
It is convenient to begin the analysis of the Laplacian on branched
coverings with the case of a single branch point, i.e., we look at a
real version of the Riemann surface of √(z). In the case of a
single branch point one can use separation of variables in polar
coordinates.  We take the liberty of using the same symbols M_0,
_0, H_0 etc. as in the case of two branch points.  For most of
our results the corresponding analogue for the case of two branch
points will be immediate; cf. a.2.

Let M_0 denote the -manifold obtained by joining two
copies of ^2 along the line {(x,0) ∈^2 | x ≤ 0} in
the usual crosswise fashion. Equipped with the Euclidean metric tensor
= (δ_ij) we obtain the Riemannian manifold _0 =
(M_0, ) with the single branch point (0,0).  The origin
(0,0) does not belong to M_0 and _0 is not complete.

For r > 0 we let B_r ⊂ M_0 denote the set of points in
M_0 with distance less than r from the origin; the “discs”
B_r form a two-sheeted covering of the punctured disc (x,y)
  ∈^20 < x^2 + y^2 < r^2.


In order to define the Laplacian H_0 of _0 we consider the
Hilbert space _0 := _0, with scalar product
denoted by ··, and the Sobolev space _0, given as the completion of M_0 with respect to
the norm [1] · defined by

    [1] ψ
        := ∫_M_0ψ(x) + ∇ψ(x) x, 
            ψ∈M_0.
  
Then H_0 is defined as the unique self-adjoint operator satisfying
(H) ⊂M_0 and

    H_0 u v 
      = ∫_M_0∇ u ·∇ v x, 
           u ∈(H_0),     v ∈M_0.

By elliptic regularity, we have (H_0) ⊂[2] M_0
and H_0 u = -Δ u for all u ∈ (H_0). More precisely, if
u belongs to (H_0), then the restriction of u to M_0
∖B_ belongs to [2]M_0 ∖B_, for any > 0. We note as an aside that
(H_0) ⊄[2]M_0. Indeed, the function u
M_0, defined in polar coordinates by u(r, θ) := 1/√(r)sin rcosθ/2, satisfies -Δ u = u in
M_0. If we now take any smooth function ϕM_0 which
is 1 on B_1 and vanishes outside of B_2, say, then ϕ u ∈(H_0) but, by a straight-forward calculation, (ϕ u)_xx∉M_0.


Another natural Sobolev space is the space M_0 = [2] 1
M_0, consisting of all functions in M_0 that have first
order distributional derivatives in M_0.  For an open set
Ω⊂^d with smooth boundary, Ω is
associated with a (weak form of) Dirichlet boundary conditions

while the Laplacian with form domain Ω is called the Neumann Laplacian of Ω.  In the case at
hand, however, the Sobolev spaces M_0 and M_0 coincide. 
For completeness, we include the (standard) proof. 



  
  We have M_0 = M_0.



  Let 0 ≤ u ∈M_0 and let u_n := min{u, n} for n
  ∈.  Then u_n → u in M_0
  (cf. <cit.>) and we see that M_0∩M_0 is dense in M_0. Consider a sequence of
  Lipschitz continuous functions ϕ_k M_0→ [0,1] with
  the following properties: ϕ_k vanishes on B_1/k and
  ϕ_k(x) = 1 for x ∉ B_2/k; furthermore, there exists a
  constant c such that |∇ϕ_k(x)| ≤ c/k, for all k ∈.  For any n ∈ fixed, we have ϕ_k u_n → u_n in
  M_0 and ∇(ϕ_k u_n) →∇ u_n weakly in
  (M_0)^2, as k →∞.  Thus, for any > 0,
  there exist n_0 ∈ and a (finite) convex combination v_
  of the ϕ_k u_n_0 such that [1]v_ - u_n_0 <. But v_∈M_0, and the result follows.




By A.1 


there is only one self-adjoint extension of the Laplacian on M_0 with form-domain contained in the Sobolev space M_0.
On the other hand, it is easy to see that H_0 is not
essentially self-adjoint on M_0. Indeed, we may just follow
the line of arguments leading to <cit.>) for
the Laplacian in ^2.  In the present situation, we use separation
of variables in polar coordinates (r,θ), with r > 0 and the
angle variable θ running through [0, 4π) instead of
[0,2π). The eigenvalues of the angular operator are now given by
κ_ℓ = - 1/4ℓ^2 with ℓ∈_0. As a
consequence, the corresponding radial operators (cf. eqns. (X.18)
in <cit.>)
 
    h_ℓ :=  -^2/ r^2 + ℓ^2 - 1/4 r^2 ,     ℓ∈_0,
 
 are not essentially self-adjoint on 0,∞  for ℓ = 0 and for ℓ = 1. 



We next consider the Rellich compactness property. In the following
lemma we let χ_r denote the characteristic function of B_r. 


  
  For all R > 0 the operators χ_R (H_0 + 1)^-1/2 and
  (H_0 + 1)^-1/2χ_R are compact.
  
  



  It is clearly enough to show that the mapping M_0∋ u
  ↦χ_R u ∈M_0 is compact.  Away from the origin
  we may apply the standard Rellich Compactness Theorem, but we need a
  different argument in a neighborhood of the origin.
  
  
  * 
    
    Let us first show that the embedding M_0↪M_0 is compact.  Indeed, any compact subset K ⊂M_0 can be covered by a finite number of discs B_r(_i),
    i=1,…, n with suitable n ∈, 0 < r < {K,
    (0,0)}, and _i ∈M_0. Then B_r(_i) ⊂M_0 and each disc B_r(_i) is (equivalent to) a Euclidean
    disc in ^2.  We may then use a partition of unity subordinate
    to this covering of K and we may apply the usual Rellich
    Compactness Theorem in each B_r(_i).

  
  * 
 Let us define the Dirichlet Laplacian H_0;1 of B_1 as the 
 (unique) self-adjoint operator with  quadratic form domain 
 B_1 and with quadratic form (<ref>). 
 Using again separation of variables in polar coordinates as above, 
 we have to deal with the Friedrichs extension of the operators  
 h_ℓ  on 0,1, for ℓ∈_0. 
 Each of the operators h_ℓ has purely discrete spectrum with the 
 lowest eigenvalue tending to ∞ as ℓ→∞. 
  It follows that H_0;1 has compact resolvent.   
   
  
  * 
    Let (u_k) ⊂M_0 and suppose that u_k → 0
    weakly in M_0.  It is enough to show that χ_R u_k →
    0 in M_0 strongly, for all R > 0.

    Choose a (smooth) cutoff-function ϕ with support in
    B_1 and which is equal to 1 in B_1/2. We then have
    ϕ u_k ∈B_1 and ϕ u_k → 0
    weakly in B_1. By the second part of this proof 
   H_0;1 has compact resolvent. This implies that ϕ u_k → 0 in
    B_1 since
    
    ϕ u_k^2 = H_0;1^-1(∇(ϕ u_k))∇(ϕ u_k)

    where ∇(ϕ u_k) → 0 weakly and
    H_0;1^-1(∇(ϕ u_k)) → 0 strongly in
    B_1/2.

    On the other hand, B_R ∖ B_1/2 is a relatively compact
    subset of M_0, and therefore (1 - ϕ) χ_R u_k → 0 in
    M_0 by part lem.1.2.i of this proof.
  




We next comment on the spectral properties of H_0. As H_0 ≥
0 we have H_0⊂ [0,∞). Clearly, [ess]
H_0⊃ [0,∞) and so H_0 = [ess] H_0
= [0,∞).  All operators h_ℓ in (<ref>)
have purely absolutely continuous spectrum since 0 is not an
eigenvalue and the operators h_ℓ are purely a.c. in
(0,∞); cf., e.g., <cit.>. It is then clear 
that H_0 is also purely a.c.; in other words, H_0 has no
singular continuous spectrum and has no eigenvalues.





 §.§ Double coverings with two branch points.




We now return to the manifold  with two branch points and the
associated Laplacian H as in wave-ops-euclid. As in the case of a single
branch point the Sobolev spaces  and 
coincide. Again, H is not essentially self-adjoint on M. Also
(H + 1)^-1χ_R is compact for all R > 0 with χ_R as in
wave-ops-euclid. 
 The proofs require only some obvious modifications as
compared to the case of a single branch point.  As for the spectral
properties of H it is again clear that H = [ess] H =
[0,∞) and it remains to deal with the question of absolute
continuity.  Here we refer to some work of
Donnelly <cit.> and Kumura <cit.> who have
pertinent statements for complete manifolds which are asymptotically
Euclidean. It is clear from their proofs that the presence of a finite
number of branch points can be accomodated.


As an alternative, it is easy to adapt the Enß method of
scattering (cf. e.g. <cit.>) to exclude singular continuous spectrum of H.
The absence of eigenvalues can be obtained as in the Kato-Agmon-Simon
theorem in <cit.>:



 
  The Laplacian H of (M,) has no eigenvalues.



  Clearly, 0 cannot be an eigenvalue of H since an eigenfunction
  for the eigenvalue 0 would have to be constant.  Positive
  eigenvalues can be excluded by following the proof of the
  Kato-Agmon-Simon Theorem <cit.> with some
  obvious modifications and simplifications.  In the case at hand, the
  operator H is not essentially self-adjoint on M, but any
  eigenfunction ψ of H is clearly in M and there is a
  sequence of smooth functions ψ_n ∈(H), vanishing outside
  the radius n, such that ψ_n →ψ and H ψ_n → H
  ψ in M, as n →∞.
  

For the present paper it is quite useful—albeit not essential—to know
that the Laplacian of (M,) is purely absolutely continuous. Of
course, it is also natural to ask whether the operators H_g on M
with metric g as in pert-metric are purely absolutely
continuous. Here the papers <cit.> and <cit.>
mentioned above give sufficient conditions. 












§ STATIONARY PHASE ESTIMATES

  



We refer to <cit.> for the basics of stationary phase
estimates.  In this appendix we consider two functions ψ_1,
ψ_2 ∈ with ψ_1 = ψ_2 = 1,
and we let u_0 := ψ_1 ⊗ψ_2 ∈^2.

We let h_0 denote the (unique) self-adjoint extension of -^2/ x^2 on  and we let A_0 denote the (unique)
self-adjoint extension of -Δ on ^2 so that A_0
= h_0 ⊗ I_{y} + I_{x}⊗ h_0.  We then write

    Ψ_i
       = Ψ_i(x,t)
       := ^- th_0ψ_i,      i = 1, 2    t ∈;

in particular, we have  

    ^- t A_0 u_0
      = (^- th_0ψ_1) ⊗  (^- th_0ψ_2) 
      = Ψ_1(·,t) ⊗Ψ_2(·,t), 
           t ∈.

We will be using the following basic estimate on the real line where
ψ̂= ψ denotes the Fourier transform for ψ∈. It is clearly enough to consider t ≥ 0, in the sequel. 


  
  Let ψ∈ with ψ = 1 and let Ψ =
  Ψ(·,t) := ^- t h_0ψ.
  
  
  * 
    Suppose ψ̂∈ and let a ≥ 0 be such that
    ψ̂⊂ [-a,a].  We then have: For any m ∈
    there exists a constant c_m ≥ 0 such that
    
    Ψ(x,t)≤ c_m (1 + |x| + t)^-m , 
               x ≥ 2at.


  
  * 
    Suppose there exists s ≥ 0 such that ψ̂⊂
    [s,s+1]. We then have: For any m ∈ there exists a
    constant c_m ≥ 0 such that
    
    Ψ(x,t)≤ c_m (1 + |x - 2st| + t )^-m, 
                x ≤ 2st,

    for all m ∈, where the constant c_m can be chosen
    independently of s.
  


a.1 is an immediate consequence of classical
stationary phase estimates, as discussed, e.g.,
in Appendix 1 to Section XI.3 of <cit.>. 
A motivation for these estimates is that
the “classically allowed” region for ^- t A_0(ψ_1 ⊗ψ_2) at time t ≥ 0 is contained in the rectangle [-2at,2at] ×
[2st, 2(s+1)t] if ψ_1 and ψ_2 are as in
a.1lem.a.1.i and lem.a.1.ii, respectively.

We use the estimates (<ref>) and (<ref>) in the
following lemma where

    Q_s,t 
      := [-st, st] × [st, ∞)  ⊂^2,
           t ≥ 0,

and χ_s,t is the characteristic function of Q_s,t. 




  
  Let u_0 = ψ_1 ⊗ψ_2 as above where ψ_1 satisfies
  the assumptions of a.1 (i) and ψ_2 satisfies the
  assumptions of a.1lem.a.1.ii.

  We then have: for any m ∈ there exists a constant c_m
  ≥ 0 such that for t ≥ 0 and s ≥ 2a
  
    (1 - χ_s,t)  ^- tA_0u_0^2 
        ≤c_m (1 + st)^1 - 2m,

  and
  
    (1 - χ_s,t)  ∇^- tA_0u_0^2 
        ≤c_m (1 + st)^1 - 2m.




  By a.1 we have
  
    ∫_|x| > st |Ψ_1(x,t)|^2  x  
        ≤ 2 c_m^2 ∫_st^∞ ( 1 + |x|)^-2m x 
        = 2 c^2_m/2m-1 ( 1 + st)^1-2m

  and, similarly,
  
    ∫_y < stΨ_2(y,t) y 
        ≤c^2_m/2m-1 ( 1 + st)^1-2m,

  for all t > 0 and s ≥ 2a.  Using Eqn. (<ref>), we therefore
  obtain
  
    (1 - χ_s,t) ^- tA_0 u_0   ≤ C_0 Ψ_2(·,t)∫_|x| >  stΨ_1(x,t) x
               + C_0 Ψ_1(·,t)∫_y < st |Ψ_2(y,t)|^2  y
       ≤ c_m (1 + st)^1-2m

  with (non-negative) constants C_0, c_m that are independent
  of s.  This proves (<ref>).  For the
  estimate (<ref>), we use the well-known fact that p := -/ x and ^- th_0 = ^- t p^2 commute,
  
  
  whence
  
    ∇^- t A_0 u_0 = (^- th_0ψ_1' ⊗ψ_2, 
        ψ_1 ⊗^- th_0ψ_2').
 
  Proceeding as above, we obtain (<ref>) with a constant depending on
  ψ_1' and ψ_2'.





We are now ready to provide the basic estimate for the “localization
error” as in eqn. (<ref>). 



  
  Suppose ψ_1 and ψ_2 are as in a.2.  In addition, let
  χ∈^2 with χ, ∇χ, Δχ
  bounded, and such that χ⊂(x,y) ∈^2 x ≤ 1 +  y.  Let f = f(x,y,t) = - 2 (∇χ)
  ·∇^- tA_0 u_0 -  (Δχ) ^- tA_0
  u_0 with u_0 = ψ_1 ⊗ψ_2. We then have: For any
  > 0 there exists s_≥ 0 such that
  
    ∫_-∞^∞f(·,·,t) t < , 
             s ≥ s_.
 


 
  Without restriction we may assume s ≥ 2a.  We only consider t >
  0, the case t < 0 being almost identical.

  The set Q_s,t = (-st, st) × (st, ∞) does not
  intersect the double cone (x,y) y <  x⊂^2 and ∇χ and Δχ vanish on Q_s,t.  Then
  the estimates (<ref>) and (<ref>) immediately imply
  that for any m ∈ there exists a constant C_m ≥ 0 such
  that
  
    f(·,·,t)≤ C_m (1 + st)^(1-2m)/2,
             s ≥ 2a.
 
  We now fix some m ≥ 2 and integrate with respect to t to
  obtain
  
    ∫_0^∞f(·,·,t) t
        ≤ 
        C_m ∫_0^∞  ( 1 + ts)^(1-2m)/2 t < ∞,
 
  where the integral on the right hand side tends to zero, as s →∞.






§ LOWER BOUNDS FOR THE INJECTIVITY RADIUS

  



Lower bounds for the injectivity radius are crucial for the
applicability of our results to concrete examples. We are now going to
explain how a comparison result of Müller and
Salomonsen <cit.> can be used to deal with
various situations where the metric is associated with the graph of a
function on ^2 or on ^2 ∖{(0,0)}.  These estimates
may be of independent interest. Appendix D of <cit.> contains
related results for radially symmetric manifolds. Let us first recall
the basic comparison result:




  
  Let M denote a smooth n-dimensional manifold.  Suppose that the
  Riemannian manifolds _0 := (M,g_0) and _1 : = (M,g_1)
  are complete with quasi-isometric metrics g_0 and g_1, i.e.,
  
    η g_0 ≤ g_1 ≤η^-1 g_0,
 
  for some constant η∈ (0,1]; cf. 2.2.  Furthermore,
  suppose that the sectional curvature of _0 and _1 is
  bounded (in absolute value) by some constant K ≥ 0. Let
  __0(p) and __1(p) denote the
  injectivity radius of _0 and _1, respectively, at the
  point p ∈ M.  We then have
  
    __1(p)
        ≥1/2min{η^2 π/√(K), 
                       η__0(p) },
              p ∈ M.
 



Note that the assumptions of b.1 are global and that the
manifolds are assumed to be complete. We will use simple cut-offs and
also an extension procedure for functions of class [2] to
obtain local versions.

In the sequel, we will deal with the special case n=2, _0 =
(^2, ) and _1 = (^2, g_f) where the metric g_f comes from
a function f ^2 → of class [2], as in
examples.  We start with the particularly simple case where the
first and second order derivatives of f are bounded.



  
  Let f ∈[2](^2, ) with bounded derivatives of the
  first and second order. Let g = g_f as defined above, and let
  _1 := (^2, g_f). If β≥ 0 and γ > 0 are constants such
  that
  
    |D_i f(p)| ≤β,      |D_ij(p)| ≤γ,

  for all p ∈^2 and i,j ∈{1,2}, then the radius of
  injectivity of _1 at p ∈^2 satisfies
  
    __1(p) 
        ≥π/2 √(2)·1/(1 + 2β^2)^2 γ, 
             p ∈^2.




  The injectivity radius of _0 = (^2, ) is infinite at all p
  ∈^2.  As for the constants η and K in b.1
  we may take η := 1/(1 + 2β^2) and K := 2 γ^2 since
  the curvature κ satisfies |κ(p)| ≤ | H_f(p)| by
  eqn. (<ref>). The desired estimate now follows from
  eqn. (<ref>).
 



Henceforth we will drop the factor π/(2√(2)) > 1 for better
readability.  We next consider f ∈[2](^2, ) without
assuming a bound for the derivatives of f.



  
  Let M = ^2 and let f ∈[2](^2, ).  Let g =
  g_f as defined above, and let _1 := (^2, g_f).  For p_0 ∈
  M, let
  
    β(p_0) 
           := max|D_i f(p)||p - p_0| ≤ 2,  i ∈{ 1,2},
    γ(p_0)
           := max|D_ij f(p)||p - p_0| ≤ 2,  i,j ∈{1,2}.

  Then there is a constant c ≥ 0, which is independent of f, such
  that the radius of injectivity of _1 at p_0 ∈^2 satisfies
  
    __1(p_0) 
        ≥min{ 1, (1 + 2c^2β(p_0)^2)^-2  
        (γ(p_0) + cβ(p_0))^-1}.




  Let p_0 ∈^2. We may assume f(p_0) = 0
  without restriction of generality since the lower bound
  of (<ref>) depends only on derivatives of f.  Let
  ϕ∈B_2(p_0) satisfy 0 ≤ϕ≤ 1 and ϕ(p) =
  1 for p ∈ B_1(p_0).  The function f̃ := ϕ f has
  support contained in B_2(p_0).  Since f(p_0) = 0 we have
  |f̃(p)| ≤ 2 √(2)β(p_0) for all p ∈ B_2(p_0) by
  the mean value theorem. Routine calculations then lead to the
  estimates
  
    |D_i f̃(p)|  ≤ c_ϕβ(p_0),      
        |D_ijf̃(p)|  ≤γ(p_0) + c_ϕβ(p_0),

  for all p ∈^2 and i,j ∈{1, 2}, where c_ϕ is a
  constant depending only on a bound for the first and second order
  derivatives of ϕ. We may also assume that these bounds are
  independent of p_0 ∈^2.  Applying the
  estimate (<ref>) with c_ϕβ(p_0) and
  γ(p_0) + c_ϕβ(p_0) replacing β and γ,
  respectively, we obtain the desired result.




In order to deal with branch points or other singularities, we now
consider functions f on the punctured plane ^2_* := ^2
∖{(0,0)}.  The method used in the proof of b.3 could
be easily adapted to the case where p_0 is close to the
origin. However, this would require working with cut-offs ϕ which
are supported in B_2 ρ(p_0) and which are equal to 1 on
B_ρ(p_0), for some 0 < ρ < 1/2 |p_0|. In this case the
constant c_ϕ in the proof of the
estimate (<ref>) would blow up like |p_0|^-2, as
p_0 → (0,0).  We therefore first restrict f to a suitable
half-disc (with positive distance to the origin) and then use a
[2]-extension method.

 


  
  For f ∈[2](^2_*, ) we define the metric g = g_f
  on ^2_* as before and we let _1 := (^2_*, g_f).  For p_0
  ∈^2_* we consider the annulus
  
    A(p_0) := 
        { p ∈^2 |1/2 |p_0| ≤  |p| ≤1/2 |p_0| + 2}
 
  and we define 
  
    β(p_0)      
        :=  max|D_i f(p)|p ∈ A(p_0),    i ∈{1,2},
    γ(p_0)     
        :=  max|D_ij f(p)|p ∈ A(p_0),  i,j ∈{1,2}.

  Then the radius of injectivity of _1 at p_0 ∈^2_* with
  |p_0| ≤ 1 satisfies the lower bound
  
    __1(p_0) 
        ≥min{ |p_0|/2, (1 + 2c^2β(p_0)^2)^-2 
               (γ(p_0) + cβ(p_0))^-1},

  where c ≥ 0 is a constant which can be chosen to be independent
  of f and p_0.
 


  Without restriction of generality we may assume that p_0 = (x_0,
  0) with 0 < x_0 ≤ 1. For the following construction we refer to
  4. We write r_0 := 1/2 x_0 and we let p_1 := (1/2 x_0, 0) = 1/2 p_0. Then the circle ∂
  B_r_0(p_0) passes through the point p_1. Let H_+(p_1) ⊂^2 denote the half-plane to the right of p_1, i.e.,
  
    H_+(p_1) := { p = (x,y) ∈^2 | 2 x >  x_0 }.

  It is easy to see that B_r_0(p_0) is contained in B_1,+(p_1)
  := H_+(p_1) ∩ B_1(p_1). Furthermore, B_2,+(p_1) := H_+(p_1)
  ∩ B_2(p_1) is contained in the annulus A(p_0).

  


  We now apply the well-known formula for the extension of a function
  of class [2] across a hyperplane as in <cit.> to obtain an extension F of f from
  the (closure of) the half-disc B_2,+(p_1) into the disc
  B_2(p_1) satisfying the following estimates, valid for all p ∈
  B_2(p_1):
 
    |D_i F(p)|    ≤ 
            C_max|D_i f(q)|q ∈ B_2,+(p_1)≤ C_β(p_0),      
    
        |D_ij F(p)|    ≤ 
            C_max|D_ij f(q)|q ∈ B_2,+(p_1)≤ C_γ(p_0),
 
 for some constant C_≥ 0 as in <cit.>.
 
 We may now proceed as in the proof of b.3: choose a cut-off
 function ϕ∈B_2(p_1) satisfying ϕ(p) = 1 for all
 p ∈ B_1(p_1) and let f̃ := ϕ F.  We then take c :=
 c_ϕ C_ with c_ϕ as in the proof of b.3,
 and the desired estimate follows as before.
 





  Higher order reflections are just one method of obtaining extensions
  of functions of class [2]. In the case of b.4 the
  geometry is particularly simple and we can use reflection at a
  line. Here the orders of differentiation are not mixed in the sense
  that the bounds for the k-th order derivatives of the extended
  function F depend solely on bounds for the k-th order
  derivatives of f, for k = 1,2.

  In a more complicated geometric setting, one could work with
  extension from the closed disc B_r_0(p_0)
  using <cit.> or employing an
  extension theorem of Whitney type as
  in <cit.>. An advantage of Whitney extension
  lies in the fact that the constant C_ can be chosen to be
  independent of the size of the disc B_r_0(p_0); on
  the other hand, Whitney extension would involve bounds on some
  Hölder-norm for the second order derivatives.



We finally return to M as in the body of the paper, with the branch
points . This is the case which is needed in Section 3. We have
the following result. 


  
  Let M be the double covering of ^2 with the branch points  
 and let f ∈[2](M, ) with bounded first and
  second order derivatives.  We define the metric g = g_f on M as
  before and we let _1 := (M, g_f).  Then there is a constant
  c_f > 0 such that the radius of injectivity of _1 at p ∈
  M satisfies the lower bound
  
    __1(p) 
        ≥ c_f min{1, (p, q_+), (p,q_-)}.

  


  If p_0 ∈ M has distance at least 2 to , the
  estimate (<ref>) applies. In the other cases we may
  proceed as in the proof of b.4 with some more or less obvious
  modifications which we indicate now:
  
  
  
  
  
  
  * Since the distance between  is 2, we need to scale
    down all sizes in the proof of b.4 by a factor smaller than
    1.

  
  * The annulus A(p_0) will now run through both sheets.
    
  
  * Since the first and second order derivatives of f are
    bounded, the numbers β(p_0) and γ(p_0) can be
    estimated uniformly by a fixed constant.
  













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