On Open Scattering Channels for a Branched Covering of the Euclidean Plane

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 arXiv:1202.0333 and that this property is stable under suitable perturbations of the metric.


Introduction
Let M denote a branched covering of the plane, obtained by glueing two copies of R 2 along a straight-line cut between the points q − = (−1, 0) and q + = (+1, 0), where the northern edge of the upper copy of R 2 is joined to the southern edge of the lower copy, and vice-versa (see Figure 1). The branch points q ± 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 g E of R 2 , we obtain a smooth, connected Riemannian manifold M = (M, g E ) with curvature zero; note, however, that M is not complete. In the second part of the paper we will consider Riemmannian metrics g on M which are close to g E in a suitable sense so that the perturbational results of [HPW14] can be applied.
We let H denote the Laplacian of M , a self-adjoint operator acting in the Hilbert space H = L 2 (M ). 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 (e −itH ; t ∈ R) and (e −itHg ; t ∈ R). 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?" ) + q + P P P P P P P P P q For the comparison dynamics (with two scattering channels) we take the free Laplacian on two copies of R 2 which we may imagine to lie one atop of the other. In other words, we consider the Hilbert space H 0 = L 2 (R 2 ) ⊕ L 2 (R 2 ) and we let H 0 denote the direct sum of two copies of the self-adjoint Laplacian in L 2 (R 2 ), where the indices and u mean "lower" and "upper," respectively. H 0 is (purely) absolutely continuous. With a natural (unitary) identification J : H 0 → H the wave operators W ± (H, H 0 , J) = s-lim t→±∞ e itH Je −itH 0 , exist, are complete, and isometric, as will be seen in Section 2. Since also H is absolutely continuous the wave operators W ± (H, H 0 , J) are in fact unitary. Writing J = J ⊕ J u , the channel wave operators W ± (H, H 0, , J ) and W ± (H, H 0,u , J u ) are given by W ± (H, H 0,k , J k ) = s-lim t→±∞ e itH J k e −itH 0,k , k ∈ { , u}.
Note that f ∈ Ran W + (H, H 0,u , J u ) means that there exists h ∈ H u such that in particular, e −itH f 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 M , or, in the terminology of [HPW14], 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 Theorem 2.6. We next ask whether the scattering channels remain open when the Euclidean metric g E on M is replaced with a more general metric g on M which is close to g E at infinity in the sense of [HPW14]. The corresponding assumptions concern, in particular, the harmonic radius [AC92,HPW14]) and the injectivity radius of (M, g), and the difference of the Riemannian metrics g E and g in a suitable distance function. Here we profit in several ways from the fact that the geometry of M is so simple. We require that the metrics g and g E be quasi-isometric in the usual sense (cf. Definition 3.2), and we assume a global bound on the curvature of (M, g). Under additional assumptions on g, expressed in terms of the distanced 1 (g E , g) in eqn. (3.6), Theorem 3.3 states that the wave operators W ± (H g , H g E , I g ) := s-lim t→±∞ e itHg I g e −itHg E (1.1) exist and are complete, where H g E = H is the Laplacian of (M, g E ), H g is the Laplacian of (M, g), and I g is the natural identification between L 2 (M, g E ) and L 2 (M, g); as was mentioned earlier, H g E is purely absolutely continuous. In Theorem 3.3, 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 g E has to satisfy a global, quantitative smallness condition. Then Theorem 3.4 establishes the strong convergence of the scattering operators to S(H, H 0 , J) for a sequence of metrics g ε on M tending to g E as ε ↓ 0. In Corollary 3.5 we then obtain the openness of all scattering channels for small ε. The paper is organized as follows. In Section 2 we introduce most of our notation and we discuss some basic spectral properties of the manifold M = (M, g E ), 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 Section 2 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 q ± at time t = 0 at high speed, and which are essentially localized to a double cone.
In Section 3 we consider metrics g on M that are close (or, at least, close at infinity) to the Euclidean metric g E . In essence, we only have to write down what the basic definitions and results of [HPW14] mean in the present context. We then find simple conditions for the existence and completeness of the wave operators (1.1) as well as for a non-trivial interaction between the scattering channels for (M, g).
The main results of Section 3 are illustrated in Section 4 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 Theorem 3.3. 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. Appendix A is devoted to self-adjoint extensions, compactness and spectral properties of the Laplacian with metrics g E and g. As for the absolute continuity of H g E and H g , we mainly refer to some work of Donnelly [Do99] and Kumura [Ku10,Ku13].
In Appendix B 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 S (R 2 ) we multiply u(t) := e it∆ u 0 by a cut-off function χ and obtain estimates for ∇χ · ∇u(t) and (∆χ)u(t) in the L 2 -norm.
Appendix C is devoted to lower bounds for the injectivity radius of (M, g) where the metric g on R 2 or on M is close to the Euclidean metric. Starting from a comparison result of Müller and Salomonsen [MSa07] we obtain "local" versions by means of cut-offs and extension theorems, proceeding from We conclude the introduction with a few remarks concerning the literature. The paper [HPW14] 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 [GTh17]. 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. [Hi10] and [FHH15]. There is a connection between the analysis of the Aharonov-Bohm effect in Quantum Mechanics and branched coverings of Euclidean space; cf. [BHO09]. 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 [IT01] 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 H is concerned we mostly follow [K66] and [RS80]. In particular, we let H ac (T ) denote the absolutely continuous subspace of H associated with T , and P ac (T ) the orthogonal projection onto H ac (T ). For the general formal setup of multichannel scattering we refer to Section 4 of [HPW14] 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 p of M by ((x, y), ) or ((x, y), u) where (x, y) ∈ R 2 and " " means "lower," "u" 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 g E denoting the metric tensor g E = (δ ij ) we obtain the Riemannian manifold M := (M, g E ). For the remainder of this section we will be cavalier about the distinction between M and M = (M, g E ) and we will mostly write M . For two points p 1 , p 2 ∈ M the (geodesic) distance is then given by 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 q − and q + . The infimum in (2.1) is attained either for a straight line segment connecting p 1 and p 2 or for (the union of) two straight line segments that meet at one of the branch points. E.g., if p 1 = ((0, y), ), p 2 = ((0, −y), ) with y > 0, then dist(p 1 , p 2 ) = dist(p 1 , q − ) + dist(q − , p 2 ) = 2 1 + y 2 (see Figure 2 left). For a point p 0 ∈ M , we denote the (geodesic) disc of radius r > 0 and center (x 0 , y 0 ) by B r (p 0 ), i.e., such discs may or may not contain points in both sheets (see Figure 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{dist(p 0 , q + ), dist(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) ∈ R 2 ; 0 < x 2 + y 2 < r 2 }. The Riemannian manifold M is not (geodesically) complete. In order to define the Laplacian H of M , we consider the Hilbert space H := L 2 (M ) with scalar product denoted by ·, · , and the Sobolev spaceH 1 (M ), given as the completion of C ∞ c (M ) with respect to the norm ||·|| 1 defined by Then H is defined as the unique self-adjoint operator satisfying Dom(H) ⊂H 1 (M ) and It is easy to see (cf. Appendix A) thatH 1 (M ) coincides with the Sobolev space H 1 (M ) = W 1 2 (M ), consisting of all functions in L 2 (M ) that have first order distributional derivatives in L 2 (M ). Hence the Laplacian on C ∞ c (M ) has only one self-adjoint extension with form domain contained in H 1 (M ). However, the Laplacian is not essentially selfadjoint on C ∞ c (M ). Basic spectral properties of H are also discussed in Appendix A; in particular, H is purely absolutely continuous with σ(H) = σ ac (H) = [0, ∞).
We next consider the Rellich compactness property. For the proof we refer to Proposition A.2 in Appendix A. Figure 2). Then the mapping H 1 (M ) u → χ R u ∈ L 2 (M ) 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 .
We denote the straight line segment in R 2 connecting the points q ± as Γ, Remark 2.3. As is often the case in two Hilbert space scattering [K67,RS79], 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 ∈ C ∞ c (R 2 ). Proof of Proposition 2.2. We decouple both H and H 0 by Dirichlet boundary conditions along two circles defined as follows. Let C 2 := { (x, y) ∈ R 2 ; x 2 + y 2 = 4 }, C 2 := C 2 × { , u} ⊂ M 0 , and C 2 := ι(C 2 ) ⊂ M . Introducing Dirichlet boundary conditions on C 2 and on C 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 R 2 , we have the Dirichlet Laplacian h int on the disc of radius 2 and the Dirichlet Laplacian h ext on the exterior of this disc. Furthermore, defining we denote by H int the Dirichlet Laplacian of M int . Note that M int is a branched covering with two sheets of the punctured disc { (x, y) ∈ R 2 ; x 2 + y 2 < 4 } \ {q + , q − }. We then write (2.11) simply act as the identity on L 2 (M ext , g E ), and as the zero operator on L 2 (M int , g E ). 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 u and H 0, as defined above, we furthermore see that the channel wave operators (2.12) (and, analogously, W ± (H, H 0,u , J u ) exist and are isometric with (2.14) in particular, e −itH f is asymptotically on the lower sheet for t → ∞. Eqn. (2.13) establishes two orthogonal decompositions of H ac (H) = L 2 (M, g E ), one for the plus-sign and another one for the minus-sign. We will see later on (cf. Lemma 2.11) that these two decompositions are in fact different.
Remark 2.4. Let us note that H 0 = H 0, ⊕ H 0,u provides a reference operator for H in the sense of [HPW14, Def. 4.7] with two channels. Strictly speaking, branch points like q ± are not directly included in the framework used in [HPW14]. However, this technical difficulty is easy to resolve: we might just take each of the sets B 1/2 (q ± ) as an end, albeit an end which does not participate in the scattering process since the Dirichlet Laplacian of B 1/2 (q ± ) has compact resolvent by Lemma 2.1. The possibility of allowing such "dead ends" is described in Remark 4.4 of [HPW14]. We thus have (formally) a manifold with 4 ends, with two ends given by a copy of R 2 \ B 2 (0) and another two ends given by It is a major goal in scattering theory to obtain information on the scattering operator a unitary operator, and the closely related scattering matrix (S ij ) i,j∈{ ,u} , with (2.16) for i, j ∈ { , u}. 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 e −itH w 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 R 2 . We then have: , and t 0 ≥ 0 such that the following estimates hold: and In the proof of Lemma 2.5 we basically construct a state v 0 ∈ L 2 (R 2 ) which passes at high speed between the points q ± under the evolution determined by e −itA 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 (e −itA 0 ; t ∈ R), acting in L 2 (R 2 ), while we know much less about (e −itH ; t ∈ R), acting in L 2 (M, g E ). 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 e −itH w 0 from the properties of e −itA 0 v 0 using the fact that both operators act locally as the Laplacian.
Recall that H 0,u and H 0, denote the self-adjoint Laplacian in L 2 (M 0,u , g E ) and in L 2 (M 0, , g E ), respectively. We let F denote the Fourier transform on the Schwartz spaces S (R d ) for d ∈ N. It is well known that F acts bijectively on S (R d ) and extends to a unitary map F : Our construction starts with a function u 0 ∈ S (R 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 ε ∈ (0, 1) be given and let ε := ε/5. We first pick a function ϕ 1 ∈ C ∞ c (R) of norm 1 and we let ψ 1 := F −1 ϕ 1 ∈ S (R) where we assume that (2.19) We let a = a ε > 0 be such that supp ϕ 1 ⊂ (−a, a). Next, let ϕ 2 ∈ C ∞ c (0, 1), of norm 1 again, and let ψ 2 := F −1 [ϕ 2 (. − s)] ∈ S (R), where s > 0 will be chosen later. Let (2.20) Then u 0 ∈ S (R 2 ) ⊂ Dom(A 0 ) and u(t) := e −itA 0 u 0 is a classical solution of the initial value problem for the Schrödinger equation in L 2 (R 2 ), i.e., We write for s > 0, and we let χ s,t denote the characteristic function of Q s,t . Lemma B.2 implies that for any m ∈ N there exists a constant c m ≥ 0 such that so that for s ≥ 2a and t large, t ≥ t 0 say, (2.21) Now let Ω := (x, y) ∈ R 2 ; |x| < 1/2 + |y| , (2.22) let χ Ω denote the characteristic function of Ω, and, finally, where (j δ ) δ>0 is the kernel of the usual Friedrichs mollifier on R 2 ; in particular, 0 ≤ j δ ∈ C ∞ c (R 2 ) with support in the closed disc of radius δ, and j δ = 1. Also let X denote the support of χ and X the characteristic function of X , i.e, X = χ X . Note that χ is independent of t. Figure 3. Left: the wave packet u0 at time 0 has speed s in y-direction and is concentrated in x-direction near x = 0; the support of the wave packet u(t) = e iA 0 u 0 at time t > 0 is essentially contained in the dark grey area Qs,t. Moreover, when considering the time evolution v(t) of the initial state v0 = χu0 (with a cut-off function χ defined as a smooth version of the indicator function χΩ) with support in X , the deviation from u(t) is small. Right: the corresponding sets and the wave packet w(t) corresponding to v(t) on M . The initial state here is w0 = w(0). We next consider v 0 := χu 0 and observe that the (smooth) function v := χu is a solution of the inhomogeneous initial value probleṁ We also have ||v 0 − u 0 || < ε and ||v 0 || > 1−ε . Stationary phase estimates (cf. Lemma B.3 in the appendix) imply that there exists s 0 ≥ 0 such that Notice that there is no reason to expect that for t = 0 the individual terms e −itA 0 v 0 and t 0 e i(t−τ )A 0 f (τ ) dτ on the right-hand side of (2.27) should vanish outside of X ; it is only the sum of the two terms which has support contained in X . It is immediate from eqn. (2.21), ||u 0 − v 0 || < ε , and ||u 0 || = 1 that (2.28) We have now gathered all the information we need on e −itA 0 v 0 and are ready for the proof of Lemma 2.5.
Proof of Lemma 2.5. (i) In order to make the transition from R 2 to M we define a map j : X → M which assigns to (x, y) ∈ X the point ((x, y), ) ∈ M for y < 0, and the point ((x, y), u) ∈ M for y > 0. The points in X 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 X := j(X ). For functions η : X → C, we obtain a lifting Jη : X → C defined by We may extend Jη by zero to all of M . Obviously, we have w(t) := Jv(t) ∈ Dom(H) for We conclude from eqns. (2.27) and (2.31) that We finally define w 0 := Jv 0 and note that ||w 0 || > 1 − ε .
It is now easy to prove the main result of this section.
Theorem 2.6. The entries of the scattering matrix (S ij ) i,j∈{ ,u} , as defined in Eqn. (2.16), are all non-zero operators.
Proof. (i) We first show that the operator S u is non-zero. Let 0 < ε < 1/4 and let v 0 and w 0 be as in Lemma 2.5. Without loss of generality we may assume, in addition, that ||w 0 || = ||v 0 || = 1. Then where, by Lemma 2.5, It now follows that | S u v 0 , v 0 − 1| ≤ 3ε < 3/4. This shows that S u is non-zero; but then, by symmetry, we also have S u = 0.
(ii) In order to show that S (and, analogously, S uu ) 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 Lemma 2.5 to achieve this goal; cf. also Remark 2.8 below. E.g., we may replace the function ψ 1 in the proof of Lemma 2.5 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 X in the x-direction accordingly. The maps j and J can be simply defined as an embedding of X into M 0, . We leave the details to the reader.
Remark 2.7. 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 ε > 0 there are states where the norm of the outgoing wave packet on the upper sheet is greater than (1 − ε) times the norm of what is coming in on the lower sheet, etc. One might say then that the channels are strongly open.
Remark 2.8. 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 X which avoids the slit and one gets the impression that "most" initial states will belong to the range of S or S uu while only a tiny fraction of initial states communicates between the two sheets under the evolution e −itH . 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).
Remark 2.9. 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.
Remark 2.10. 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 Lemma 2.5 with some further basic properties of w 0 .
Lemma 2.11. Let P ±,u and P ±, denote the projections onto the ranges of the wave operators W ± (H, H 0,u , J u ) and W ± (H, H 0, , J ), respectively. For ε > 0 let w 0 be as in Lemma 2.5. We then have: and Proof. We only show (2.34); the proof of (2.35) is analogous and ommitted. By the Projection Theorem, we have (2.37) In order to obtain a lower bound on ||P +,u w 0 || we choose ϕ := v 0 in Eqn. (2.36) and use Lemma 2.5 to find For an upper bound on ||P +, w 0 || we use J u J = 0, combined with Lemma 2.5, to see that ||P +, w 0 || < ε.
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 (2.39) Let us first show that the adjoints (W ± (H, H 0,u , J u )) * of the wave operators W ± (H, H 0,u , J u ) are given by strong limits, exist. Here H 0 = H 0, ⊕ H 0,u and J * = (P , P u ) and we see that e itH 0 J * e −itH = e itH 0, P e −itH , e itH 0,u P u e −itH . (2.42) The ranges of e itH 0, P e −itH and e itH 0,u P u e −itH being orthogonal, it is clear that the strong limit of the left hand side of (2.42) can only exist if the strong limits of both terms on the right hand side exist (as t → ±∞).

Perturbations of the Metric
We first recall some notions and definitions in Differential Geometry as used in [HPW14]. Given a (smooth) Riemannian metric g = (g ij ) on the C ∞ -manifold M , we denote by M = (M, g) the Riemannian manifold and we let B δ (p) = B δ,M (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 [HPW14] 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 M (p) at p ∈ M is defined as in [AC92]  We denote by L 2 (M ) the usual space of (equivalence classes of) L 2 -integrable functions on the Riemannian manifold M = (M, g) with respect to the Riemannian measure d vol g . The following definition is standard.
Definition 3.2 (cf. [HPW14, Def. 3.1]). We say that the Riemannian metrics g 1 , g 2 are quasi-isometric if there exists a constant η > 0 such that for all ξ ∈ T M and p ∈ M .
In our case T M can be identified with R 2 . The Hilbert spaces L 2 (M 1 ) and L 2 (M 2 ) coincide if M 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 ∈ L 2 (M 1 ) to the same function f in L 2 (M 2 ).
We now take a closer look at the property that the metrics g and g E are quasi-isometric. Let A(p) be the endomorphism on T M given by g(p)(ξ, ξ) = g E (A(p)ξ, ξ) for all ξ ∈ T p M 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 p M in the standard coordinates, then the α k (p) are also the eigenvalues of (g ij (p)). Thus g and g E 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 [HPW14,eqns. (3.2) and (3.5)] d(g E , g)(p) := max k α k (p) 1/2 − α k (p) −1/2 , (3.5) d ∞ (g E , g) := sup p∈M d(g E , g)(p) and d 1 (g E , g) := M d(g E , g)(p)r 0 (p) −4 dp. (3.6) We call d 1 (g E , g) the weighted L 1 -quasi-distance of g and g E ; we have dropped the symmetrizing factor 1 + g E ,g (p) of d 1 appearing in [HPW14, eqn. (3.5)] (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 g E (this is equivalent with d ∞ (g, g E ) < ∞), and denote by M = (M, g) the corresponding Riemannian manifold. Then there is a (unique) self-adjoint Laplacian H g , acting in the Hilbert space L 2 (M ), with quadratic form domain given by the Sobolev spaceH 1 (M ), and defined by for any u ∈ Dom(H g ) ⊂H 1 (M ) and v ∈H 1 (M ), where (g ij ) is the inverse of (g ij ). In the Euclidean case (g = g E ) the operator H g E agrees with the operator H defined in Section 2; recall that H g E is purely a.c. From Theorem 3.7 of [HPW14] we now obtain the following result on the existence and completeness of the wave operators.
Theorem 3.3. Suppose we are given a continuous function r 0 : M −→ (0, 1] satisfying condition (2.3) and a metric g ∈ Met r 0 (M ) which is quasi-isometric to the Euclidean metric g E on M . We also assume that the difference between g and g E satisfies the r 0 -dependent weighted integral condition d 1 (g, g E ) < ∞ with d 1 as in (3.6).
Remark. Under suitable conditions on g the operator H g will be absolutely continuous (cf. Donnelly [Do99], Kumura [Ku10,Ku13]. In this case the wave operators in (3.8) are even unitary.
Remark. In applying the fundamental perturbation theorems in [HPW14] we can deal with the branch points q ± in the way described in Remark 2.4, i.e., we have (formally) a manifold with four ends, with two ends given by M ext as in Eqn. (2.7) and two ends given by B 1/2 (q ± ). Again, the ends B 1/2 (q ± ) do not participate in the scattering.
Following the development in Section 5 of [HPW14] 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 [HPW14] we define for r 0 as above and γ, ε > 0 i.e., Met r 0 (M, g E , γ, ε) is the set of smooth metrics g on M enjoying the following properties: (i) 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. (ii) The metric g is quasi-isometric to g E with the bound d ∞ (g, g E ) ≤ γ.
(iii) The weighted L 1 -quasi-distance d 1 (g, g E ) is not larger than ε. Note that condition (iii) requires a quantitative smallness of the deviation of g from the Euclidean metric in the sense that d 1 (g, g E ) ≤ ε while the main assumption in Theorem 3.3 only stipulates d 1 (g, g E ) < ∞.
Then Theorem 5.1 of [HPW14] yields the strong convergence of the scattering operators as ε ↓ 0, and Cor. 5.3 of [HPW14] establishes the openness of the scattering channels, for small ε > 0. We are now going to make this precise.
Let γ > 0 be fixed. For ε > 0, we consider g ε ∈ Met r 0 (M, g E , γ, ε) and we let H gε denote the Laplacian of (M, g ε ). The natural identification operator from L 2 (M, g E ) to L 2 (M, g ε ) is written I gε . Then the scattering operator is given by with H 0 and J as in Section 2, Proposition 2.2, and the scattering matrix (S ij ) i,j∈{ ,u} is defined by Corollary 3.5. For any γ > 0 fixed, there exists ε 0 > 0 such that S ik (H gε , H 0 , I ε J) = 0 for all metrics g ε ∈ Met r 0 (M, g E , γ, ε) and all 0 < ε ≤ ε 0 .

Examples
We first illustrate Theorem 3.3 in the special case where the perturbed metric on M is associated with the graph of a function f : M → R of class C 2 . As usual, we define Φ : M → R 3 by Φ(p) := (p, f (p)) and where J Φ is the Jacobian of Φ. The eigenvalues of g are 1 and det g = 1 + f 2 x + f 2 y . The curvature κ of M := (M, g) is given by the well-known formula where the distances are measured in (M, g E ). We have the following proposition. Then the wave operators W ± (H g , H g E , J) exist and are complete.
Proof. Since g = g f has the eigenvalues 1 and 1 + f 2 x + f 2 y 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 Met 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. (3.3) the simplest choice appears to be r 0 := d 0 with a constant ∈ (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. (3.2) is satisfied provided ≤ 1/ √ K. According to Proposition C.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 > 0 satisfying ≤ min{1/2, 1/ √ K, c 0 }. In remains to show that d 1 (g E , g) as in eqn. (2.6) is finite. Here we first estimate therefore condition (4.3) implies d 1 (g E , g) < ∞. Since the assumptions of Theorem 3.3 are satisfied, we may conclude that the wave operators for the pair (H g E , H g ) exist and are complete.
(ii) 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 → R 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 (4.3).
In an analogous way one can indicate simple conditions on f which allow the application of Theorem 3.4. We consider functions f : M → R of class C 2 with first and second order derivatives bounded by some constant C and which are such that g f ∈ Met r 0 (M ) with r 0 as above. Then d ∞ (g f , g E ) ≤ C 2 , and we may now choose γ := C 2 . For ε > 0, the condition in this case, we have g f ∈ Met r 0 (M, g E , γ, ε) and the results of Theorem 3.4 and Corollary 3.5 apply.
Proposition 4.3. Suppose we are given a sequence (f n ) ⊂ C 2 (M ) enjoying the following properties: (i) 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 ∈ N. (ii) We have M |∇f n | 2 d −4 0 dp → 0, n → ∞. (4.5) 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 gn , H 0 , I n J) exist and converge strongly to S(H g E , H 0 , J), as n → ∞.
Appendix A. Self-Adjointness and Spectral Properties.
In this appendix we study the Sobolev spacesH 1 and Laplace-Beltrami operators on branched coverings of the Euclidean plane. Here we are mainly interested in selfadjointness, compactness properties, and the question of absolute continuity of the Laplacian.
A.1. 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 , M 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. Section A.2.
Let M 0 denote the C ∞ -manifold obtained by joining two copies of R 2 along the line {(x, 0) ∈ R 2 ; x ≤ 0} in the usual crosswise fashion. Equipped with the Euclidean metric tensor g E = (δ ij ) we obtain the Riemannian manifold M 0 = (M 0 , g E ) with the single branch point (0, 0). The origin (0, 0) does not belong to M 0 and M 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) ∈ R 2 ; 0 < x 2 + y 2 < r 2 }.
In order to define the Laplacian H 0 of M 0 we consider the Hilbert space H 0 := L 2 (M 0 ), with scalar product denoted by ·, · , and the Sobolev spaceH 1 (M 0 ), given as the completion of C ∞ c (M 0 ) with respect to the norm ||·|| 1 defined by Then H 0 is defined as the unique self-adjoint operator satisfying Dom(H) ⊂H 1 (M 0 ) and By elliptic regularity, we have Dom(H 0 ) ⊂ H 2 loc (M 0 ) and H 0 u = −∆u for all u ∈ Dom(H 0 ). More precisely, if u belongs to Dom(H 0 ), then the restriction of u to M 0 \ B ε belongs to H 2 (M 0 \ B ε ), for any ε > 0. We note as an aside that Dom(H 0 ) ⊂ H 2 (M 0 ). Indeed, the function u : M 0 −→ R, defined in polar coordinates by u(r, ϑ) := 1 √ r sin r cos ϑ 2 , satisfies −∆u = u in M 0 . If we now take any smooth function ϕ : M 0 −→ R which is 1 on B 1 and vanishes outside of B 2 , say, then ϕu ∈ Dom(H 0 ) but, by a straight-forward calculation, (ϕu) xx / ∈ L 2 (M 0 ). Another natural Sobolev space is the space H 1 (M 0 ) = W 1 2 (M 0 ), consisting of all functions in L 2 (M 0 ) that have first order distributional derivatives in L 2 (M 0 ). For an open set Ω ⊂ R d with smooth boundary,H 1 (Ω) is associated with a (weak form of) Dirichlet boundary conditions while the Laplacian with form domain H 1 (Ω) is called the Neumann Laplacian of Ω. In the case at hand, however, the Sobolev spaces H 1 (M 0 ) andH 1 (M 0 ) coincide. For completeness, we include the (standard) proof.
Proof. Let 0 ≤ u ∈ H 1 (M 0 ) and let u n := min{u, n} for n ∈ N. Then u n → u in H 1 (M 0 ) (cf. [GT83]) and we see that . 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 ∈ N. For any n ∈ N fixed, we have ϕ k u n → u n in L 2 (M 0 ) and ∇(ϕ k u n ) → ∇u n weakly in (L 2 (M 0 )) 2 , as k → ∞. Thus, for any ε > 0, there exist n 0 ∈ N and a (finite) convex combination v ε of the ϕ k u n 0 such that ||v ε − u n 0 || 1 < ε.
But v ε ∈H 1 (M 0 ), and the result follows.
By Lemma A.1 there is only one self-adjoint extension of the Laplacian on C ∞ c (M 0 ) with form-domain contained in the Sobolev space H 1 (M 0 ). On the other hand, it is easy to see that H 0 is not essentially self-adjoint on C ∞ c (M 0 ). Indeed, we may just follow the line of arguments leading to [RS75, Thm. X.11]) for the Laplacian in R 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 = − 1 4 2 with ∈ N 0 . As a consequence, the corresponding radial operators (cf. eqns. (X.18) in [RS75,loc. cit.]) are not essentially self-adjoint on C ∞ c (0, ∞) for = 0 and for = 1. We next consider the Rellich compactness property. In the following lemma we let χ r denote the characteristic function of B r .
Proof. It is clearly enough to show that the mapping H 1 (M 0 ) u → χ R u ∈ L 2 (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.
(i) Let us first show that the embedding H 1 (M 0 ) → L 2,loc (M 0 ) is compact. Indeed, any compact subset K ⊂ M 0 can be covered by a finite number of discs B r (p i ), i = 1, . . . , n with suitable n ∈ N, 0 < r < dist{K, (0, 0)}, and p i ∈ M 0 . Then B r (p i ) ⊂ M 0 and each disc B r (p i ) is (equivalent to) a Euclidean disc in R 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 (p i ).
(ii) Let us define the Dirichlet Laplacian H 0;1 of B 1 as the (unique) self-adjoint operator with quadratic form domainH 1 (B 1 ) and with quadratic form (A.1). Using again separation of variables in polar coordinates as above, we have to deal with the Friedrichs extension of the operators h on C ∞ c (0, 1), for ∈ N 0 . Each of the operators h has purely discrete spectrum with the lowest eigenvalue tending to ∞ as → ∞. It follows that H 0;1 has compact resolvent.
(iii) Let (u k ) ⊂H 1 (M 0 ) and suppose that u k → 0 weakly in H 1 (M 0 ). It is enough to show that χ R u k → 0 in L 2 (M 0 ) strongly, for all R > 0.
A.2. Double coverings with two branch points. We now return to the manifold M with two branch points and the associated Laplacian H as in Section 2. As in the case of a single branch point the Sobolev spaces H 1 (M ) andH 1 (M ) coincide. Again, H is not essentially self-adjoint on C ∞ c (M ). Also (H + 1) −1 χ R is compact for all R > 0 with χ R as in Section 2. 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 σ(H) = σ ess (H) = [0, ∞) and it remains to deal with the question of absolute continuity.
Here we refer to some work of Donnelly [Do99] and Kumura [Ku10,Ku13] 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. [RS79]) to exclude singular continuous spectrum of H. The absence of eigenvalues can be obtained as in the Kato-Agmon-Simon theorem in [RS78]: The Laplacian H of (M, g E ) has no eigenvalues.
Proof. 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 [RS78, Thm. XIII.58] with some obvious modifications and simplifications. In the case at hand, the operator H is not essentially self-adjoint on C ∞ c (M ), but any eigenfunction ψ of H is clearly in C ∞ (M ) and there is a sequence of smooth functions ψ n ∈ Dom(H), vanishing outside the radius n, such that ψ n → ψ and Hψ n → Hψ in L 2 (M ), as n → ∞.
For the present paper it is quite useful-albeit not essential-to know that the Laplacian of (M, g E ) is purely absolutely continuous. Of course, it is also natural to ask whether the operators H g on M with metric g as in Section 3 are purely absolutely continuous. Here the papers [Do99] and [Ku10,Ku13] mentioned above give sufficient conditions.

Appendix B. Stationary Phase Estimates
We refer to [RS79] for the basics of stationary phase estimates. In this appendix we consider two functions ψ 1 , ψ 2 ∈ S (R) with ||ψ 1 || = ||ψ 2 || = 1, and we let u 0 := ψ 1 ⊗ ψ 2 ∈ S (R 2 ). We let h 0 denote the (unique) self-adjoint extension of − d 2 dx 2 on C ∞ c (R) and we let A 0 denote the (unique) self-adjoint extension of −∆ on C ∞ c (R 2 ) so that A 0 = h 0 ⊗ I {y} + I {x} ⊗ h 0 . We then write in particular, we have We will be using the following basic estimate on the real line whereψ = F ψ denotes the Fourier transform for ψ ∈ S (R). It is clearly enough to consider t ≥ 0, in the sequel. Lemma B.1 is an immediate consequence of classical stationary phase estimates, as discussed, e.g., in Appendix 1 to Section XI.3 of [RS79]. A motivation for these estimates is that the "classically allowed" region for e −itA 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 Lemma B.1 (i) and (ii), respectively.
We use the estimates (B.1) and (B.2) in the following lemma where and χ s,t is the characteristic function of Q s,t .
We are now ready to provide the basic estimate for the "localization error" as in eqn. (2.26).
We now fix some m ≥ 2 and integrate with respect to t to obtain where the integral on the right hand side tends to zero, as s → ∞.

Appendix C. 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 [MSa07] can be used to deal with various situations where the metric is associated with the graph of a function on R 2 or on R 2 \ {(0, 0)}. These estimates may be of independent interest. Appendix D of [HPW14] contains related results for radially symmetric manifolds. Let us first recall the basic comparison result: Note that the assumptions of Proposition C.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 C 2 to obtain local versions.
In the sequel, we will deal with the special case n = 2, M 0 = (R 2 , g E ) and M 1 = (R 2 , g f ) where the metric g f comes from a function f : R 2 → R of class C 2 , as in Section 4. We start with the particularly simple case where the first and second order derivatives of f are bounded.
We now apply the well-known formula for the extension of a function of class C 2 across a hyperplane as in [GT83, Lemma 6.37] 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 ext max |D i f (q)| ; q ∈ B 2,+ (p 1 ) ≤ C ext β(p 0 ), |D ij F (p)| ≤ C ext max |D ij f (q)| ; q ∈ B 2,+ (p 1 ) ≤ C ext γ(p 0 ), for some constant C ext ≥ 0 as in [GT83,loc. cit.]. We may now proceed as in the proof of Proposition C.3: choose a cut-off function ϕ ∈ C ∞ c (B 2 (p 1 )) satisfying ϕ(p) = 1 for all p ∈ B 1 (p 1 ) and letf := ϕF . We then take c := c ϕ C ext with c ϕ as in the proof of Proposition C.3, and the desired estimate follows as before.
Remark. Higher order reflections are just one method of obtaining extensions of functions of class C 2 . In the case of Proposition C.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 [GT83, Lemma 6.37] or employing an extension theorem of Whitney type as in [St70, Sec. VI.2.3]. An advantage of Whitney extension lies in the fact that the constant C ext 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 q ± . This is the case which is needed in Section 3. We have the following result.
Proposition C.5. Let M be the double covering of R 2 with the branch points q ± and let f ∈ C 2 (M, R) with bounded first and second order derivatives. We define the metric g = g f on M as before and we let M 1 := (M, g f ). Then there is a constant c f > 0 such that the radius of injectivity of M 1 at p ∈ M satisfies the lower bound inj M 1 (p) ≥ c f min{1, dist(p, q + ), dist(p, q − )}. (C.7) Proof. If p 0 ∈ M has distance at least 2 to q ± , the estimate (C.5) applies. In the other cases we may proceed as in the proof of Proposition C.4 with some more or less obvious modifications which we indicate now: (i) Since the distance between q ± is 2, we need to scale down all sizes in the proof of Proposition C.4 by a factor smaller than 1. (ii) The annulus A(p 0 ) will now run through both sheets. (iii) 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.