On the Scattering Theory of the Laplacian with a Periodic Boundary Condition. II. Additional Channels of Scattering

We study spectral and scattering properties of the Lapla- cian H (¾) = i¢ in L2(R 2 ) corresponding to the boundary condition @u @o +¾u = 0 for a wide class of periodic functions ¾. For non-negative ¾ we prove that H (¾) is unitarily equivalent to the Neumann Lapla- cian H (0) . In general, there appear additional channels of scattering


Introduction 0.1 Setting of the problem
The present paper is a continuation of [Fr], but can be read independently.It studies the Laplacian together with a boundary condition of the third type where ν denotes the exterior unit normal and where the function σ : R → R is assumed to be 2π-periodic.Moreover, let σ ∈ L q,loc (R) for some q > 1.
Under this condition H (σ) can be defined as a self-adjoint operator in L 2 (R 2 + ) by means of the lower semibounded and closed quadratic form We analyze the spectrum of H (σ) and develop a scattering theory viewing H (σ)  as a (rather singular) perturbation of H (0) , the Neumann Laplacian on R 2 + .(For the abstract mathematical scattering theory see, e.g., [Ya1].)By means of the Bloch-Floquet theory we represent H (σ) as a direct integral with fiber operators H (σ) (k) acting in L 2 (Π) where Π := (−π, π) × R + is the halfstrip.Due to the relation (0.3) the investigation of the operator H (σ)  reduces to the study of the operators H (σ) (k).

The main results
It was shown in [Fr] that the wave operators on the halfstrip exist and are complete.This immediately implies the existence of the wave operators W (σ) ± := W ± (H (σ) , H (0) ) on the halfplane and the coincidence of the ranges (Of course, the existence of the wave operators can also be obtained by a modification of the Cook method, see Section 17 in [Ya2].)Moreover, it was shown in [Fr] that the singular continuous spectrum of the operators H (σ) (k) is empty.
In the present paper we will study the point spectrum of the operators H (σ) (k).In general, there will be (discrete or embedded) eigenvalues which may produce bands in the spectrum of the operator H (σ) on the halfplane.In this case, the wave operators are not complete and there appear additional channels of scattering.For the additional bands in the spectrum we give some quantitative estimates and we construct an example where a gap in the spectrum appears.Moreover, we prove that the spectrum of the operator H (σ) is purely absolutely continuous.Under the additional assumption σ(x 1 ) ≥ 0, a.e.x 1 ∈ R, (0.4) we prove that the operators H (σ) (k) have no eigenvalues.This implies that the wave operators W (σ) ± are unitary and provide a unitary equivalence between the operators H (σ) and H (0) .

Additional channels of scattering
Additional channels of scattering were already discovered in a number of other problems that exhibit periodicity with respect to some but not all space directions.Without aiming at completeness we mention the papers [DaSi], [Sa] concerning the scattering theory of problems of this type, [GrHoMe], [Ka] concerning Schrödinger operators with periodic point interactions and [BeBrPa] concerning the case of discrete Schrödinger operators.In the present paper, using the specific properties of the operator under consideration we are able not only to show the appearance of additional channels of scattering but also to develop a more detailed analysis of these channels.In particular, we give some sufficient conditions for existence and non-existence of additional channels and prove that the spectrum of the operator is purely absolutely continuous.The problem of absolute continuity in a case with partial periodicity is also investigated in [FiKl], where the Schrödinger operator with an electric potential is considered.

Outline of the paper
Let us explain the structure of this paper.In Section 1 we recall the precise definition of the operators H (σ) and H (σ) (k) in terms of quadratic forms and the direct integral decomposition.In Subsection 1.2 we state the main result in the case of non-negative σ (Theorem 1.1) and the main result about absolute continuity (Theorem 1.2).In Section 2 we transform the eigenvalue problem for H (σ) (k) and λ ∈ R in the spirit of the Birman-Schwinger principle to the problem whether 0 is an eigenvalue of a certain "discrete pseudo-differential operator" of order one in L 2 (T).In this way we reduce the problem of (possibly embedded) eigenvalues to the study of operators with compact resolvent.In Section 3 we prove the absence of eigenvalues of H (σ) (k) under the condition (0.4), which implies Theorem 1.1.The general case is treated in Section 4 and the proof of Theorem 1.2 is given in Subsection 4.3.We supplement this in Section 5 with a more detailed analysis in the case when σ is a trigonometric polynomial.Finally, in Section 6 we describe and discuss the additional channels of scattering that appear in the general case.In Subsection 6.2 we construct an example of an open gap.

Acknowledgements
The authors are deeply grateful to Prof. M. Sh.Birman for the setting of the problem, useful discussions and constant attention to the work.The authors also thank Prof. T. A. Suslina for useful consultations and Prof. N. Filonov and Prof. F. Klopp for making us [FiKl] accessible before its publication.The authors thank Prof. A. Laptev, the KTH Stockholm, the Mittag-Leffler Institute as well as Prof. H. Siedentop and the LMU Munich for hospitality.The first author acknowledges gratefully the partial financial support of the German Merit Foundation and of the European Union through the IHP network HPRN-CT-2002-00277.The second author acknowledges gratefully the partial financial support of RFBR (grant no.02-01-00798) and of the German Academic Exchange Service through the IQN network.
1 Setting of the problem.The main result

Notation
We introduce the halfplane and the halfstrip where R + := (0, +∞).Moreover, we need the lattice 2πZ.Unless stated otherwise, periodicity conditions are understood with repect to this lattice.We think of the corresponding torus T := R/2πZ as the interval [−π, π] with endpoints identified.We use the notation For a measurable set Λ ⊂ R we denote by meas Λ its Lebesgue measure.
For an open set Ω ⊂ R d , d = 1, 2, the index in the notation of the norm .L2(Ω) is usually dropped.The space L 2 (T) may be formally identified with L 2 (−π, π).We denote the Fourier coefficients of a function f ∈ L 2 (T) by fn := 1 is the Sobolev space of order s ∈ R (with integrability index 2).By H s (T) we denote the closure of C ∞ (T) in H s (−π, π).Here C ∞ (T) is the space of functions in C ∞ (−π, π) which can be extended 2π-periodically to functions in C ∞ (R).The space H s (T) is endowed with the norm By Hs (Π) we denote the closure of C∞ (Π) ∩ H s (Π) in H s (Π).Here C∞ (Π) is the space of functions in C ∞ (Π) which can be extended 2π-periodically with respect to x 1 to functions in C ∞ (R 2 + ).Statements and formulae which contain the double index "±" are understood as two independent assertions.

1.2
The operators H (σ) on the halfplane.Main results Before describing the main results we recall the definition of the operators H (σ) from [Fr].Let σ be a real-valued periodic function satisfying for some q > 1. (1.1) It is easy to see (cf.[Fr]) that under this condition the quadratic form is lower semibounded and closed in the Hilbert space L 2 (R 2 + ), so it generates a self-adjoint operator which will be denoted by H (σ) .The case σ = 0 corresponds to the Neumann Laplacian on the halfplane, whereas the case σ = 0 implements a (generalized) boundary condition of the third type.The spectrum of the "unperturbed" operator H (0) coincides with [0, +∞) and is purely absolutely continuous of infinite multiplicity.In [Fr] we proved the existence of the wave operators We state now the main results of the present part.An especially complete result can be obtained under the additional assumption σ(x 1 ) ≥ 0, a.e.x 1 ∈ R. (1.3) Theorem 1.1.Assume that σ satisfies (1.1) and (1.3).Then the wave operators W (σ) ± exist, are unitary and satisfy In particular, under the condition (1.3) the spectrum of the operator H (σ) is purely absolutely continuous.This is also true for general σ.
However, in contrast to the case of non-negative σ now the operator H (σ) may be not unitarily equivalent to H (0) and then the wave operators W (σ) ± are not complete.This is connected with the existence of additional channels of scattering.The discussion of this phenomenon is conveniently postponed to Section 6.Let σ be a real-valued periodic function satisfying (1.1) and let k ∈ [− 1 2 , 1 2 ].It follows (cf.[Fr]) that the quadratic form
The operator H (σ) on the halfplane can be partially diagonalized by means of the Gelfand transformation.This operator is initially defined for u ∈ S(R 2 + ), the Schwartz class on R 2 + , by and extended by continuity to a unitary operator ⊕L 2 (Π) dk. (1.6) One finds (cf.[Fr]) that This relation allows us to investigate the operator H (σ) by studying the fibers In [Fr] it was shown that In the present part we give a detailed analysis of the point spectrum of 2 Characterization of eigenvalues of the operator Let σ be a real-valued periodic function satisfying (1.1) and let k ∈ [− 1 2 , 1 2 ], λ ∈ R. In the Hilbert space L 2 (T) we consider the quadratic forms (2.1) where It follows from the Sobolev embedding theorems that the forms b (σ) (λ, k) are lower semibounded and closed, so they generate self-adjoint operators which will be denoted by The compactness of the embedding of H1/2 (T) in L 2 (T) implies that the operators B (σ) (λ, k) have compact resolvent.Now we characterize the eigenvalues of the operator H (σ) (k) as the values λ for which 0 is an eigenvalue of the operators B (σ) (λ, k).More precisely, we have Then u ∈ N (H (σ) (k) − λI) and, moreover, (2.3) holds.
For the proof of Proposition 2.1 we use the following notation.For u ∈ L 2 (Π) and n ∈ Z we define so that, with respect to convergence in L 2 (Π), The proof of the following observation is straightforward.
2. Let f ∈ H 1/2 (T), then the following are equivalent: Proof of Proposition 2.1.The proof follows easily from Lemma 2.2.Note that with f defined by (2.3).
Remark 2.3.Obviously, the statement of Proposition 2.1 does not depend on the definition of β n (λ, k) for (n + k) 2 ≤ λ.The reason for our choice (2.2) is of technical nature and will become clear in Subsection 4.2 below.
3 The case of non-negative σ Proposition 2.1 allows us to deduce easily the main result if σ is non-negative.
We start with the operators H (σ) (k) on the halfstrip.
Proof.In view of (1.8) it suffices to prove that H (σ) (k) has no eigenvalues.For this we use Proposition 2 Concerning the operator H (σ) on the halfplane we obtain immediately the Proof of Theorem 1.1.In [Fr] we showed that W (σ) ± is unitarily equivalent to the direct integral of the operators W (σ) empty) consists of eigenvalues of finite multiplicities which may accumulate at +∞ only.
Note that the case of an infinite sequence of (embedded) eigenvalues actually occurs.
Example 4.2.Let σ ≡ σ 0 < 0 be a negative constant and k Then This follows easily by Proposition 2.1 or directly by separation of variables.
For the proof of Theorem 4.1 we need an auxiliary result.For k ∈ [− 1 2 , 1 2 ], λ ∈ R we denote by µ m (λ, k), m ∈ N, the eigenvalues of B (σ) (λ, k) arranged in non-decreasing order and repeated according to their multiplicities.Then we have then the functions µ m (., k), m ∈ N, are continuous and strictly decreasing on R.
The proof (of strict monotonicity) uses an analyticity argument and is conveniently postponed to Subsection 4.2.
The equality (4.3) can be used to obtain estimates on the number of eigenvalues of H (σ) (k) below k 2 and on its asymptotics in the limit of large coupling constant.Such calculations for the operators B (σ) (k 2 , k) are rather standard, so we do not go into details.

Complexification
Now we extend the operator family B (σ) (λ, k) to complex values of λ and k.
From this construction we obtain Proof.The proof is rather standard, so we only sketch the major steps.We consider the family B (σ) (z, κ), z ∈ Ũ , κ ∈ Ṽ , constructed above.Since these operators have compact resolvent, we can use a Riesz projection to separate the eigenvalues around 0 from the rest of the spectrum.The resulting operator has finite rank and is analytic with respect to z and κ, so its determinant h has the desired properties.
Our next goal is to show that for every λ ∈ U the function h(λ, .)constructed above is not identically zero.For the proof of this we need to consider quasimomenta κ = k + iy with large imaginary part y.
As an immediate consequence of Lemmas 4.5 and 4.7 and relation (4.1) we obtain the following result which will be needed in Subsection 4.3 to prove that the spectrum of the operator H (σ) is purely absolutely continuous.
Corollary 4.8.There exists a countable number of open intervals U j , V j ⊂ R and real-analytic functions h j : there is a j such that (λ, k) ∈ U j × V j and h j (λ, k) = 0, and 2. for all j and all λ ∈ U j one has h j (λ, .)≡ 0.
To complete this subsection we prove Lemma 4.3 which was used in the proof of Theorem 4.1.

Proof of Theorem 1.2
Now we prove Theorem 1.2 following the method suggested in [FiKl].We need the following result from Complex Analysis of Several Variables which can be proved by means of the Implicit Function Theorem (see [FiKl]).
Lemma 4.9.Let U, V ⊂ R be open intervals and h : U ×V → C be real-analytic.Let Λ ⊂ U with meas Λ = 0 such that for all λ ∈ Λ one has h(λ, .)≡ 0. Then and we have to prove that this operator is equal to 0. For this we write [− 1 2 , 1 2 ] = K 1 ∪ K 2 ∪ K 3 where and with the notation of Corollary 4.8 It follows from Lemma 4.9 that meas K 2 = meas K 3 = 0, which concludes the proof.
We have seen in Example 4.2 that the operators H (σ) (k) may have embedded eigenvalues.Let us investigate this phenomenon under the additional assumption that only finitely many Fourier coefficients of σ are non-zero.Note that in this case the operator B (σ) (λ, k) acts in Fourier space as a finite-diagonal matrix.This allows us to exclude the existence of large embedded eigenvalues.
Proof.The proof of only if σ coincides a.e. with a negative constant, which is excluded by the assumption N > 0. Let us prove now that σ p H In particular, we see from (5.2) The estimate and (5.1) imply that fn = 0 for at least 2N consecutive n.Using σN = σ−N = 0 it is easy to see from (5.2) that fn = 0 for all n, i.e. f = 0.So by Proposition 2.1 (1), λ is not an eigenvalue of H (σ) (k).
Documenta Mathematica 9 (2004) 57-77 It follows easily from the monotonicity properties mentioned above that for each pair (m, m ) ∈ N × N there exists at most one pair (λ, k) ) such that (5.6) holds.Since the functions ν m are strictly positive for sufficiently large m we conclude that the set (5.3) is finite.
Remark 5.5.Let us mention that the eigenvalue in the above example is due to the following symmetry.Since the operator is (up to unitary equivalence) invariant under a shift with respect to x 1 we may assume that β ∈ R. Then σ is even with respect to x 1 = 0 and so for k = 0 the decomposition into even and odd functions reduces the operator H (σ) (0).It remains to notice that the essential spectrum of the part of the operator acting on odd functions starts at the point λ = 1.
6 Additional Channels of Scattering of the operators H (σ)

Additional Channels due to discrete eigenvalues
Here we construct the additional channels of scattering of H (σ) which arise from the discrete eigenvalues of the operators the discrete eigenvalues of H (σ) (k), arranged in non-decreasing order and repeated according to their multiplicities.By Theorem 4.1 l(k) is a finite number, possibly equal to 0. It is convenient to set λ l (k) := k 2 if l > l(k).The functions λ l are continuous on [− 1 2 , 1 2 ] for each l ∈ N. Combining this with (1.7) we find i.e., the spectrum of H (σ) has band structure.
According to Theorem 1.2 none of the functions λ l is constant (since this would correspond to an eigenvalue of H (σ) ).
To construct the additional channels of scattering we introduce some notation.We put These sets are open in [− 1 2 , 1 2 ] and K l = ∅ for sufficiently large l.We define Now assume l 0 > 0 (which means that some of the operators H (σ) (k) have discrete eigenvalues).For each k ∈ [− 1 2 , 1 2 ] we can choose orthonormal eigenfunctions ψ l (., k), 1 ≤ l ≤ l(k), corresponding to the eigenvalues (6.1), such that the mappings are piecewise analytic.Recall that the functions ψ l (., k) are of the form (2.4).
It is convenient to define ψ l (., k) := 0 if k ∈ K l and to extend the functions ψ l (., k) periodically with respect to the variable x 1 to functions on R ], the projection in L 2 (Π) onto the subspace spanned by ψ l (., k).With this notation, we call the subspaces additional channels of scattering (ACS) of the operator H (σ) .Here U is the Gelfand transformation (1.6).Thus the functions u ∈ C l are precisely the functions of the form In particular, it follows from the form (2.4) of the eigenfunction ψ l (., k) that functions u ∈ C l decay exponentially with respect to the variable x 2 provided K l = [− 1 2 , 1 2 ].Let us list some more properties of the spaces C l .One has for all 1 ≤ l, j ≤ l 0 C l ⊥ C j , j = l, and Indeed, this follows from the fact that ψ l (., k) is orthogonal to ψ j (., k), j = l, and to the subspace R(W In particular, Theorem 1.2 implies that the wave operators W (σ) ± are not complete if there exists an ACS (i.e., l 0 > 0).Moreover, the spaces C l reduce the operator H (σ) , and on functions u ∈ C l of the form (6.3) H (σ) acts by multiplying the function f with the function λ l .Thus, the part of H (σ) on C l is unitarily equivalent to multiplication with the function λ l on L 2 (K l ).Remark 1.10 of [Fr] shows that functions u ∈ C l correspond to states which propagate along the boundary.

Existence of ACS. Existence of gaps
It is clear from Theorem 1.1 that there are no ACS if σ is non-negative.Let us give an easy sufficient condition for the existence of ACS.It requires σ to be "negative in mean".Proposition 6.1.Assume that σ0 := Proof.Indeed, we prove that H (σ) (k) has an eigenvalue smaller or equal to . For this we consider the trial function defined by u(x) := e σ0x2/ √ 2π , x ∈ Π, which satisfies The assertion follows now from the variational principle.
Remark 6.2.With more elaborate techniques one can show that the conclusion of Proposition 6.1 remains valid under the assumption σ0 = 0, σ ≡ 0.
Remark 6.4.It follows from (6.6) that the condition λ D l < λ N l+1 for some l ∈ N is sufficient for an open gap.This can be used to construct further examples.Remark 6.5.By an argument similar to the one in Example 6.3 we find that if there exists a non-empty connected open subset Λ of the torus such that σ(x 1 ) ≤ − π meas Λ , x 1 ∈ Λ, then σ H (σ) ∩ (−∞, 0) = ∅, so there exist ACS.To conclude this subsection we note that the number of ACS (due to discrete eigenvalues) can be estimated using (4.3).

Additional Channels due to embedded eigenvalues
In general, the embedded eigenvalues of the operators H (σ) (k), k ∈ [− 1 2 , 1 2 ], also contribute to the spectrum of the operator H (σ) .Therefore the subspace σ) σ p H (σ) (k) ∩ k 2 , +∞ , k dk U may be non-trivial and, in this case, will be called an ACS.We have The subspace C * reduces the operator H (σ) and is orthogonal to the ACS C l , 1 ≤ l ≤ l 0 , and to R(W (σ) ± ).Let us consider some examples.If σ ≡ σ 0 < 0 is a negative constant, we know from Example 4.2 that the embedded eigenvalues of H (σ) (k) depend piecewise analytically on k and all of them contribute to the spectrum of H (σ) .We note that in this case the part of H (σ) on C * is an unbounded operator.If σ is a trigonometric polynomial of degree N > 0, we know from Proposition 5.1 that H (σ) (k) has no embedded eigenvalues greater or equal to (N − |k|) 2 .Moreover, we know from Proposition 5.2 that the embedded eigenvalues in the interval [(N − 1 + |k|) 2 , (N − |k|) 2 ) do not contribute to the spectrum of the operator H (σ) .So the part of H (σ) on C * is a bounded operator with spectrum contained in [0, (N − 1 2 ) 2 ].In the special case when σ is a trigonometric polynomial of degree one, it follows again from Proposition 5.1 and Proposition 5.2 that C * = {0}.We emphasize (see Example 5.4) that embedded eigenvalues of the operators H (σ) (k) actually occur in this case.The question whether C * can be non-trivial for non-constant σ remains open.Documenta Mathematica 9 (2004) 57-77 Mathematica 9 (2004) 57-77 1.3 Definition of the operators H (σ) (k) on the halfstrip.Direct Integral Decomposition The latter were shown to be complete, and by Theorem 3.1 they are actually unitary.Thus W (σ) ± is unitary and (1.4) follows from the intertwining property of wave operators.Documenta Mathematica 9 (2004) 57-77 4 The general case 4.1 The point spectrum of the operators H (σ) (k) If we impose no additional condition on σ we have the following result on the point spectrum of the operators H (σ) (k).Theorem 4.1.Assume that σ satisfies (1.1) and let