On the scattering theory of the Laplacian

with a periodic boundary condition.

I. Existence of wave operators

Rupert L. Frank

Received: December 12, 2003

Communicated by Heinz Siedentop

Abstract. We study spectral and scattering properties of the Laplacian H^{(σ)} = −Δ in L_{2}(ℝ_{+}^{2}) corresponding to the boundary condition \frac{∂u}{∂ν} + σu = 0 for a wide class of periodic functions σ. The Floquet decomposition leads to problems on an unbounded cell which are analyzed in detail. We prove that the wave operators W_{±}(H^{(σ)}, H^{(0)}) exist.

2000 Mathematics Subject Classification: Primary 35J10; Secondary 35J25, 35P05, 35P25. Keywords and Phrases: Scattering theory, periodic operator, Schrödinger operator, singular potential.

Introduction

0.1 Setting of the problem

The present paper studies the Laplacian

H^{(σ)}u = −Δu  on ℝ_{+}^{2}  (0.1)

on the halfplane together with a boundary condition of the third type

\frac{∂u}{∂ν} + σu = 0  on ℝ × \{0\},  (0.2)

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 closed and lower semibounded quadratic form

∫_{ℝ_{+}^{2}} |∇u(x)|^{2} dx + ∫_{ℝ} σ(x_{1})|u(x_{1}, 0)|^{2} dx_{1}, u ∈ H^{1}(ℝ_{+}^{2}).

This is the first part of a paper where 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., [Ya].) The main result of the present paper is that the wave operators

W_{±}^{(σ)} := W_{±}(H^{(σ)}, H^{(0)})

exist.

0.2 Physical interpretation

In the physical interpretation, H^{(σ)} is the Hamiltonian of a two-dimensional quantum-mechanical system which consists of a particle in the upper halfplane and a crystal that fills the lower halfplane. The particle can not enter the crystal but interacts non-trivially with the surface of the crystal, described by the function σ. The existence of the wave operators means that every particle which is described by a state u ∈ R (W_{±}^{(σ)}) behaves like a free particle in the distant future and the distant past. We emphasize that there may also exist

particles which are described by a state u ∈ R (W_{±}^{(σ)})^{⊥}. These are surface states which propagate along the boundary and decay exponentially away from the boundary. Such surface states will be investigated in the second part [FrSh] of the paper.

0.3 Outline of the paper

Let us explain some of the mathematical ideas involved. A precise definition of the operator H^{(σ)} in terms of a quadratic form is given in Subsection 1.4. By means of the Bloch-Floquet theory we represent H^{(σ)} in Subsection 2.2 as a direct integral

∫_{−1/2}^{1/2} ⨁ H^{(σ)}(k) dk

with fiber operators H^{(σ)}(k) acting in L_{2}(Π) where Π := (−π, π) × ℝ_{+} is the halfstrip. Functions in the domain of H^{(σ)}(k) satisfy the third type condition (0.2) on (−π, π) × {0} (at least if σ is smooth), so H^{(σ)}(k) differs from H^{(0)}(k) by a relatively compact form perturbation. This makes a rather detailed analysis of the operators H^{(σ)}(k) possible. Our approach leans on a quadratic form version of the resolvent identity which we present in Subsection 3.2 following [Ya]. A similar approach has been successfully applied to study periodic Schrödinger operators (cf. [BShSu]). In our case it allows to show that the difference of resolvents of H^{(σ)}(k) and H^{(0)}(k) belongs to the trace class, and from the Birman-Kreĭn theorem (which is sometimes called Birman-Kuroda theorem, unaware of [BKr]) we deduce in Subsection 3.4 the existence and completeness of the wave operators on the halfstrip. Using the same representation we can prove a limiting absorption principle in Subsection 3.6, which implies the absence of singular continuous spectrum. The existence of the wave operators W_{±}^{(σ)} on the halfplane is derived from the existence of the wave operators on the halfstrip.

0.4 Acknowledgements

The author is deeply grateful to Prof. Birman for the setting of the problem, useful discussions and constant attention to the work. He would like to thank Dr. R. G. Shterenberg for numerous consultations and the help in the preparation of the present paper. He is also deeply grateful to Prof. T. A. Suslina for a systematic help with the theory of embedding theorems and boundary value problems. The author appreciates the hospitality of the KTH Stockholm and the Mittag-Leffler Institute where the results were obtained. Partial financial support of the German Merit Foundation and of the European Union through the IHP network HPRN-CT-2002-00277 is gratefully acknowledged.

1 Setting of the problem. The main result

1.1 Notation

We introduce the halfplane

ℝ_{+}^{2} := {x = (x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} > 0} = ℝ × ℝ_{+},

and the halfstrip

Π := {x = (x_{1}, x_{2}) ∈ ℝ_{+}^{2} : −π < x_{1} < π, x_{2} > 0} = (−π, π) × ℝ_{+},

where ℝ_{+} := (0, +∞). Moreover, we need the lattice 2πℤ. Unless stated otherwise, periodicity conditions are understood with respect to this lattice. We think of the corresponding torus 𝕋 := ℝ/2πℤ as the interval [−π, π] with endpoints identified. We use the notation D = (D_{1}, D_{2}) = −i∇ in ℝ^{2}. For an open set Ω ⊂ ℝ^{d}, d = 1, 2, the index in the notation of the norm ∥.∥_{L_{2}(Ω)} is usually dropped. The space L_{2}(𝕋) may be formally identified with L_{2}(−π, π). We define the (discrete) Fourier transformation F : L_{2}(𝕋) → l_{2}(ℤ) by

(Ff)_{n} = f̂_{n} := \frac{1}{\sqrt{2π}} ∫_{−π}^{π} f(x_{1})e^{−inx_{1}} dx_{1}. n ∈ ℤ,

Next, for an open set Ω ⊂ ℝ^{d}, d = 1, 2, H^{s}(Ω) 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

∥f∥_{H^{s}(𝕋)}^{2} := ∑_{n∈ℤ}(1 + n^{2})^{s}|f̂_{n}|^{2}, f ∈ H^{s}(𝕋).

By H̃^{s}(Π) we denote the closure of C̃^{∞}(Π) ∩ H^{s}(Π) in H^{s}(Π). Here C̃^{∞}(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 Scattering theory

Here we summarize the definitions and basic results on scattering theory. For proofs we refer to [Ya]. Let H_{0}, H be self-adjoint operators in a Hilbert space ℌ. The projection onto the absolutely continuous subspace of H_{0} and the unitary group of H_{0} are denoted by P_{0} and U_{0}(t) := exp(−itH_{0}), respectively. We put ℌ_{0}^{(ac)} := R(P_{0}). For the similar objects related to the operator H we omit the index ”0”. In case of existence, the limit

W_{±}(H, H_{0}) := s − lim_{t→±∞} U(−t)U_{0}(t)P_{0}

is called the wave operator for the pair H, H_{0} and the sign ±. Thus the elements u = W_{±}(H, H_{0})u_{0}^{±} ∈ R(W_{±}(H, H_{0})), u_{0}^{±} ∈ ℌ_{0}^{(ac)}, satisfy

lim_{t→±∞} ∥U(t)u − U_{0}(t)u_{0}^{±}∥ = 0.

The wave operators are partial isometries with initial subspace ℌ_{0}^{(ac)}. One easily establishes the intertwining property

W_{±}(H, H_{0}) H_{0} = H W_{±}(H, H_{0}).

It follows that the subspace R(W_{±}(H, H_{0})) and its orthogonal complement are invariant under H and that the wave operator provides a unitary equivalence between the part of H on R(W_{±}(H, H_{0})) and the absolutely continuous part of H_{0}. In particular,

R(W_{±}(H, H_{0})) ⊂ ℌ^{(ac)}.  (1.1)

The wave operator W_{±}(H, H_{0}) is said to be complete if equality holds in (1.1). It is easy to see that the completeness of W_{±}(H, H_{0}) is equivalent to the existence of W_{±}(H_{0}, H). Thus, if the wave operator W_{±}(H, H_{0}) exists and is complete, then the absolutely continuous parts of H_{0} and H are unitarily equivalent. Let us conclude this brief overview with a convenient sufficient condition for the existence and completeness of the wave operators due to Birman and Kreĭn (cf. [BKr]).

Proposition 1.1. Let H_{0}, H be self-adjoint operators on a Hilbert space ℌ such that (H − zI)^{−1} − (H_{0} − zI)^{−1} belongs to the trace class for some z ∈ ρ(H_{0}) ∩ ρ(H). Then the wave operators W_{±}(H, H_{0}) exist and are complete.

1.3 Multiplication on the boundary

Here we present auxiliary statements related to Sobolev embedding theorems. Let σ be a periodic function satisfying

σ ∈ L_{q}(𝕋)  for some q > 1.  (1.2)

It follows from the compactness of the embedding H^{1/2}(−π, π) ⊂ L_{2q′}(−π, π) (with \frac{1}{q} + \frac{1}{q′} = 1) that the form ∫_{−π}^{π} |σ(x_{1})||f(x_{1})|^{2} dx_{1}, f ∈ H^{1/2}(−π, π), is compact in H^{1/2}(−π, π). This implies

Lemma 1.2. Assume (1.2) and let ϵ > 0. Then there exists a constant C_{1}(ϵ, σ) > 0 such that

∫_{−π}^{π} |σ(x_{1})||f(x_{1})|^{2} dx_{1} ≤ ϵ∥f∥_{H^{1/2}(−π,π)}^{2} + C_{1}(ϵ, σ)∥f∥^{2}, f ∈ H^{1/2}(−π, π).

Now let us pass to the situation on the halfstrip and on the halfplane. The trace operator u ↦ u(., 0) is bounded from H^{1}(Π) to H^{1/2}(−π, π). Hence the form ∫_{−π}^{π} |σ(x_{1})||u(x_{1}, 0)|^{2} dx_{1}, u ∈ H^{1}(Π), is compact in H^{1}(Π) and we obtain

Lemma 1.3. Assume (1.2) and let ϵ > 0. Then there exists a constant C_{2}(ϵ, σ) > 0 such that

∫_{−π}^{π} |σ(x_{1})||u(x_{1}, 0)|^{2} dx_{1} ≤ ϵ∥u∥_{H^{1}(Π)}^{2} + C_{2}(ϵ, σ)∥u∥^{2}, u ∈ H^{1}(Π).

Now let u ∈ H^{1}(ℝ_{+}^{2}). For each n ∈ ℤ we apply Lemma 1.3 to the function Π ∋ x ↦ u(x_{1} + 2πn, x_{2}) and then sum over all n ∈ ℤ. This yields

Lemma 1.4. Assume (1.2) and let ϵ > 0. Then there exists a constant C_{2}(ϵ, σ) > 0 such that

∫_{ℝ} |σ(x_{1})||u(x_{1}, 0)|^{2} dx_{1} ≤ ϵ∥u∥_{H^{1}(ℝ_{+}^{2})}^{2} + C_{2}(ϵ, σ)∥u∥^{2}, u ∈ H^{1}(ℝ_{+}^{2}).

Our treatment in Section 3 needs a more precise, quantitative result on the embedding H^{1/2}(−π, π) ⊂ L_{2q′}(−π, π). We begin by recalling the definition of the weak l_{p}-spaces

l_{p,w}(ℤ) = {(α_{n})_{n∈ℤ} : sup_{t>0} tρ_{α}^{1/p}(t) < ∞}, 0 < p < ∞,

where ρ_{α}(t) := ♯{n ∈ ℤ : |α_{n}| > t} for t > 0. l_{p,w}(ℕ) is defined in a similar way. Further, recall (cf. Section 11.6 in [BS]) that Σ_{p}(ℌ_{1}, ℌ_{2}), 0 < p < ∞, is the class of compact operators K from a Hilbert space ℌ_{1} to a Hilbert space ℌ_{2} for which (s_{n}(K))_{n∈ℕ} ∈ l_{p,w}(ℕ), where (s_{n}(K))_{n∈ℕ} is the sequence of singular numbers of K. One puts Σ_{p}(ℌ_{1}) := Σ_{p}(ℌ_{1}, ℌ_{1}). The dependence on ℌ_{1}, ℌ_{2} is usually dropped in the notation if this does not lead to confusion. The connection with the well-known Schatten class 𝔖_{p} of order p (which consists of compact operators K for which (s_{n}(K))_{n∈ℕ} ∈ l_{p}(ℕ)) can be seen from the inclusions

Σ_{r} ⊂ 𝔖_{p} ⊂ Σ_{p}, r < p.

We will often use the fact that K_{1} ∈ Σ_{p_{1}}, K_{2} ∈ Σ_{p_{2}} implies

K_{1}K_{2} ∈ Σ_{( \frac{1}{p_{1}} + \frac{1}{p_{2}} )^{−1}}.  (1.3)

For a more general statement as well as for the proof of the following Cwikeltype estimate we refer to Theorem 4.8 in [BKaS].

Proposition 1.5. Let β ∈ L_{p}(𝕋) and α ∈ l_{p,w}(ℤ) for some p > 2. Then βF^{∗}α ∈ Σ_{p}(l_{2}(ℤ), L_{2}(𝕋)).

Let us consider the sequence α given by

α_{n} := (1 + n^{2})^{−1/4},  n ∈ ℤ,  (1.4)

and a function β as above. We write βF^{∗}α = (βF^{∗}α^{2/p}) α^{(p−2)/p}. Clearly, the operator of multiplication by α^{(p−2)/p} belongs to Σ_{2p/(p−2)}(l_{2}(ℤ)), and by Proposition 1.5 βF^{∗}α^{2/p} ∈ Σ_{p}(l_{2}(ℤ), L_{2}(𝕋)). Thus, taking into account (1.3), we obtain

Corollary 1.6. Let β ∈ L_{p}(𝕋) for some p > 2 and α be given by (1.4). Then βF^{∗}α ∈ Σ_{2}(l_{2}(ℤ), L_{2}(𝕋)).

This is the desired embedding result. Note that F^{∗}αF maps L_{2}(𝕋) unitarily onto H^{1/2}(𝕋).

1.4 Definition of the operators H^{(σ)} on the halfplane

Let σ be a real-valued periodic function satisfying (1.2). In the Hilbert space L_{2}(ℝ_{+}^{2}) we consider the quadratic form

\begin{aligned}D[h^{(σ)}] & := H^{1}(ℝ_{+}^{2}), \\ h^{(σ)}[u] & := ∫_{ℝ_{+}^{2}} |Du(x)|^{2} dx + ∫_{ℝ} σ(x_{1})|u(x_{1}, 0)|^{2} dx_{1}. \end{aligned}  (1.5)

According to Lemma 1.4 the form h^{(σ)} is lower semibounded and closed, so it generates a self-adjoint operator which will be denoted by H^{(σ)}. By construction

(H^{(σ)}u, v) = h^{(σ)}[u, v], u ∈ D(H^{(σ)}), v ∈ H^{1}(ℝ_{+}^{2}),

so it follows that the distributional Laplacian Δu of u ∈ D(H^{(σ)}) belongs to L_{2}(ℝ_{+}^{2}) and that

H^{(σ)}u = −Δu.

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. More precisely, we have Remark 1.7. Under the condition that σ is absolutely continuous with σ′ ∈ L_{q}(𝕋) for some q > 1 it can be proved that

D(H^{(σ)}) = \{u ∈ H^{2}(ℝ_{+}^{2}) : −\frac{∂u}{∂x_{2}} + σu = 0 on ℝ × \{0\}\} .

1.5 Main result

First we remark that the spectrum of the ”unperturbed” operator H^{(0)} coincides with [0, +∞) and is purely absolutely continuous of infinite multiplicity. This can be seen easily by applying a Fourier transformation with respect to the variable x_{1} and a Fourier cosine transformation with respect to the variable x_{2}. We turn now to the ”perturbed” operator H^{(σ)}. We have recalled the abstract definition of the wave operators in Subsection 1.2. In case of existence we will use the notation

W_{±}^{(σ)} := W_{±}(H^{(σ)}, H^{(0)}).

The main result of the present paper is

Theorem 1.8. Assume that σ satisfies (1.2). Then the wave operators W_{±}^{(σ)} exist and satisfy R(W_{+}^{(σ)}) = R(W_{−}^{(σ)}).

We note that the equality R(W_{+}^{(σ)}) = R(W_{−}^{(σ)}) implies the unitarity of the scattering matrix. It follows from Theorem 1.8 that the part of H^{(σ)} on R(W_{±}^{(σ)}) is unitarily equivalent to H^{(0)}, and so σ_{ac} (H^{(σ)}) ⊃ [0, +∞). In the second part [FrSh] we supplement this theorem with the following results. The operator H^{(σ)} has purely absolutely continuous spectrum. In general, σ (H^{(σ)}) may contain (apart from [0 + ∞)) additional bands, so the wave operators W_{±}^{(σ)} may be not complete. The spectral subspaces corresponding to the additional bands of H^{(σ)} are additional channels of scattering. However, under the additional assumption

σ ≥ 0 a.e.  (1.6)

the wave operators W_{±}^{(σ)} turn out to be complete and unitary.

Remark 1.9. The statement of Theorem 1.8 holds with obvious changes when the role of the comparison operator H_{0} is played by the Dirichlet Laplacian on the halfplane or the Laplacian on the whole plane. This follows easily from the chain rule for wave operators and the fact, that the wave operators W_{±}(H^{(0)}, H_{0}) exist and are complete. Indeed, they can be calculated easily. (Clearly, if H_{0} is the Laplacian on the whole plane one has to use an identification operator.)

Remark 1.10. A time-dependent characterization of the range of the wave operators and its orthogonal complement can be established by standard methods (see [DaSi], [Sa]): With the notation U^{(σ)}(t) := exp(−itH^{(σ)}) for t ∈ ℝ it follows that

R(W_{±}^{(σ)}) = {u ∈ L_{2}(ℝ_{+}^{2}) : lim_{t→±∞} ∫_{ℝ×(0,a)} |U^{(σ)}(t)u(x)|^{2} dx = 0, a ∈ ℝ_{+}},
R(W_{±}^{(σ)})^{⊥} = {u ∈ L_{2}(ℝ_{+}^{2}) : lim_{a→+∞} sup_{t∈ℝ} ∫_{ℝ×(a,+∞)} |U^{(σ)}(t)u(x)|^{2} dx = 0}.

Thus, the additional channels of scattering correspond to ”surface states”, i.e., states concentrated near the boundary for all time.

2 Direct integral decomposition

2.1 Definition of the operators H^{(σ)}(k) on the halfstrip

Let σ be a real-valued periodic function satisfying (1.2) and let k ∈ [−\frac{1}{2}, \frac{1}{2}]. In the Hilbert space L_{2}(Π) we consider the quadratic form

\begin{aligned}D[h^{(σ)}(k)] & := H̃^{1}(Π), \\ h^{(σ)}(k)[u] & := ∫_{Π}(|(D_{1} + k)u(x)|^{2} + |D_{2}u(x)|^{2}) dx +∫_{−π}^{π}σ(x_{1})|u(x_{1}, 0)|^{2} dx_{1}. \end{aligned} ; (2.1)

According to Lemma 1.3 the form h^{(σ)}(k) is lower semibounded and closed, so it generates a self-adjoint operator which will be denoted by H^{(σ)}(k). Similarly as above one finds

H^{(σ)}u = (D_{1} + k)^{2}u + D_{2}^{2}u = −Δu + 2kD_{1}u + k^{2}u, u ∈ D(H^{(σ)}).

In addition to the Neumann (if σ = 0) or third type (if σ ≠ 0) boundary condition at {x_{2} = 0} the functions in D(H^{(σ)}) satisfy periodic boundary conditions at {x_{1} = ±π}. A statement analogous to Remark 1.7 holds.

2.2 Direct integral decomposition of the operator H^{(σ)}

The operator H^{(σ)} can be partially diagonalized by means of the Gelfand transformation U. This operator is initially defined for u ∈ S(ℝ_{+}^{2}), the Schwartz class on ℝ_{+}^{2}, by

(Uu)(k, x) := ∑_{n∈ℤ} e^{−ik(x_{1}+2πn)}u(x_{1} + 2πn, x_{2}), k ∈ [−\frac{1}{2}, \frac{1}{2}], x ∈ Π,

and extended by continuity to a unitary operator

U : L_{2}(ℝ_{+}^{2}) → ∫_{−1/2}^{1/2} ⊕L_{2}(Π) dk.

Moreover, it turns out that u ∈ H^{1}(ℝ_{+}^{2}) iff (Uu)(k, .) ∈ H̃^{1}(Π) for a.e. k ∈ [−\frac{1}{2}, \frac{1}{2}] and ∫_{−1/2}^{1/2} ∥(Uu)(k, .)∥_{H^{1}(Π)}^{2} dk < ∞, and in this case

(UD_{1}u)(k, .) = (D_{1} + k)(Uu)(k, .), (UD_{2}u)(k, .) = D_{2}(Uu)(k, .).

Concerning the multiplication on the boundary by a periodic function σ satisfying (1.2), one finds for u ∈ H^{1}(ℝ_{+}^{2})

∫_{−1/2}^{1/2} ∫_{−π}^{π} σ(x_{1})|(Uu)(k, x_{1}, 0)|^{2} dx_{1} dk = ∫_{ℝ} σ(x_{1})|u(x_{1}, 0)|^{2} dx_{1}.

To summarize, the Gelfand transformation satisfies

U (D[h^{(σ)}]) = \{F ∈ ∫_{−1/2}^{1/2} ⊕L_{2}(Π) dk : F (k) ∈ H̃^{1}(Π) for a.e. k ∈ [−\frac{1}{2}, \frac{1}{2}],

∫_{−1/2}^{1/2} |h^{(σ)}(k)[F (k)]| dk < ∞} ,

h^{(σ)}[u] = ∫_{−1/2}^{1/2} h^{(σ)}(k)[(Uu)(k, .)] dk,  u ∈ H^{1}(ℝ_{+}^{2}),

which implies

U H^{(σ)} U^{∗} = ∫_{−1/2}^{1/2} ⊕H^{(σ)}(k) dk.  (2.2)

This relation allows us to investigate the operator H^{(σ)} by studying the fibers H^{(σ)}(k).

2.3 Main result for the operators H^{(σ)}(k) on the halfstrip

In case of existence we will use the notation

W_{±}^{(σ)}(k) := W_{±}(H^{(σ)}(k), H^{(0)}(k)), k ∈ [−\frac{1}{2}, \frac{1}{2}].

Theorem 2.1. Assume that σ satisfies (1.2) and let k ∈ [−\frac{1}{2}, \frac{1}{2}]. Then the wave operators W_{±}^{(σ)}(k) exist and are complete.

The spectrum of the ”unperturbed” operator H^{(0)}(k) (see Subsection 3.1) coincides with [k^{2}, +∞) and is purely absolutely continuous. By the remarks in Subsection 1.2 Theorem 2.1 implies that the absolutely continuous part of H^{(σ)}(k) is unitarily equivalent to H^{(0)}(k), in particular

σ_{ac} (H^{(σ)}(k)) = [k^{2}, +∞).

Concerning the singular continuous spectrum of H^{(σ)}(k) we prove

Theorem 2.2. Assume that σ satisfies (1.2) and let k ∈ [−\frac{1}{2}, \frac{1}{2}]. Then

σ_{sc} (H^{(σ)}(k)) = ∅.

The point spectrum of H^{(σ)}(k) is investigated in the second part [FrSh]. We prove there that σ_{p} (H^{(σ)}(k)) consists of eigenvalues of finite multiplicities which may accumulate at +∞ only. The situation of infinitely many (embedded) eigenvalues does actually occur. The discrete eigenvalues of H^{(σ)}(k) produce bands in the spectrum of H^{(σ)}. In general, the same is true for the embedded eigenvalues of H^{(σ)}(k). However, under the additional assumption (1.6) we prove σ_{p} (H^{(σ)}(k)) = ∅, which implies the completeness and even unitarity of the wave operators W_{±}^{(σ)}.

2.4 Reduction of Theorem 1.8 to Theorem 2.1

Assuming Theorem 2.1, the proof of Theorem 1.8 is easy.

Proof of Theorem 1.8. For t ∈ ℝ, k ∈ [−\frac{1}{2}, \frac{1}{2}] put U^{(σ)}(t) := exp(−itH^{(σ)}) and U^{(σ)}(t, k) := exp(−itH^{(σ)}(k)), and similarly with σ replaced by 0. The wave operators W_{±}^{(σ)}(k) exist by Theorem 2.1 and are measurable with respect to k, so ∫_{−1/2}^{1/2} ⊕W_{±}^{(σ)}(k) dk is well defined. Moreover, by (2.2) together with Theorem XIII.85 in [ReSi4] one has

U U^{(σ)}(t) U^{∗} = exp(−it U H^{(σ)} U^{∗}) = ∫_{−1/2}^{1/2} ⊕U^{(σ)}(t, k) dk,

where U is the Gelfand transformation from Subsection 2.2, and similarly with σ replaced by 0. It follows that for all u ∈ L_{2}(ℝ_{+}^{2})

∥ U U^{(σ)}(−t)U^{(0)}(t)u − (∫_{−1/2}^{1/2} ⊕W_{±}^{(σ)}(k) dk) Uu∥^{2} =

= ∫_{−1/2}^{1/2} ∥U^{(σ)}(−t, k)U^{(0)}(t, k) (Uu)(k, .) − W_{±}^{(σ)}(k) (Uu)(k, .)∥^{2} dk → 0

(t → ±∞)

by Lebesgue’s theorem. This means that the strong limits of U^{(σ)}(−t)U^{(0)}(t) for t → ±∞ exist and coincide with W_{±}^{(σ)} = U^{∗} (∫_{−1/2}^{1/2} ⊕W_{±}^{(σ)}(k) dk) U. In

particular, because of the completeness of W_{±}^{(σ)}(k),

R(W_{±}^{(σ)}) = U^{∗} (∫_{−1/2}^{1/2} ⊕R(P_{ac}^{(σ)}(k)) dk) U.

3 The operators H^{(σ)}(k) on the halfstrip

3.1 The unperturbed operator H^{(0)}(k) on the halfstrip

We start our investigation by summarizing results on the ”unperturbed” operator H^{(0)}(k). Let k ∈ [−\frac{1}{2}, \frac{1}{2}]. By separation of variables one easily finds that the spectrum of the operator H^{(0)}(k) coincides with [k^{2}, +∞), is purely absolutely continuous and that the spectral multiplicity of λ ∈ [k^{2}, +∞) is ♯{n ∈ ℤ : (n + k)^{2} ≤ λ}. Note that the spectral multiplicity changes at the ”threshold points” (n + k)^{2}, n ∈ ℤ. It is also easy to verify that the resolvent

R^{(0)}(z, k) := (H^{(0)}(k) − zI)^{−1} , z ∈ ρ (H^{(0)}(k)) = ℂ \ [k^{2}, +∞),

is an integral operator with kernel

\begin{matrix}r^{(0)}(x, y; z, k) := \frac{1}{4π} ∑_{n∈ℤ} \frac{e^{in(x_{1}−y_{1})}}{β_{n}(z, k)} (e^{−β_{n}(z,k) (x_{2}+y_{2})} + e^{−β_{n}(z,k) |x_{2}−y_{2}|}) , \\ x, y ∈ Π, x_{2} ≠ y_{2}. \end{matrix} ; (3.1)

where

β_{n}(z, k) := \sqrt{(n + k)^{2} − z},  n ∈ ℤ, z ∈ ℂ \ [k^{2}, +∞).  (3.2)

Here and in the following we choose the canonical branch of the square root on ℂ \ (−∞, 0] satisfying Re \sqrt{.} > 0. Note that the RHS of (3.1) converges absolutely and uniformly on compact subsets of {(x, y) ∈ Π × Π : x_{2} ≠ y_{2}}.

3.2 A general approach to the inversion of a perturbed operator

To investigate the ”perturbed” operators H^{(σ)}(k) we use a version of the resolvent identity. The ”classical” resolvent identity (H − zI)^{−1} − (H_{0} − zI)^{−1} = −(H_{0} − zI)^{−1}(H − H_{0})(H − zI)^{−1} involves the difference of H and H_{0}, which may be not well-defined if the operators are defined via quadratic forms. Here we present a version of the resolvent identity that works also in the quadratic form case. The proof may be found in Section 1.9 of [Ya]. We work in a general setting: Let ℌ be a Hilbert space and H, H_{0} be self-adjoint operators satisfying

D(|H|^{1/2}) = D(|H_{0}|^{1/2}).  (3.3)

We note that H and H_{0} are not assumed to be semibounded (but they will be so in our application). Denote their resolvents by

R_{0}(z) := (H_{0} − zI)^{−1}, z ∈ ρ(H_{0}), R(z) := (H − zI)^{−1}, z ∈ ρ(H).

Suppose that there is an ”auxiliary” Hilbert space 𝔊 and operators

G_{0} : ℌ ⊃ D(G_{0}) → 𝔊, G : ℌ ⊃ D(G) → 𝔊,

such that the following is true.

(H1) The operators G_{0}, G are |H_{0}|^{1/2}-bounded, i.e.,

\begin{matrix}D(|H_{0}|^{1/2}) ⊂ D(G_{0}), & D(|H_{0}|^{1/2}) ⊂ D(G), \\ G_{0}(|H_{0}|^{1/2} + I)^{−1} ∈ 𝔅(ℌ, 𝔊), & G(|H_{0}|^{1/2} + I)^{−1} ∈ 𝔅(ℌ, 𝔊). \end{matrix}

(H2) The operators G_{0}, G satisfy

(G_{0}f, Gg) = (Gf, G_{0}g), f, g ∈ D(|H_{0}|^{1/2}).

(H3) The relation H = H_{0} + G^{∗}G_{0} holds in the sense of forms, i.e.,

(Hf, f_{0}) = (f, H_{0}f_{0}) + (Gf, G_{0}f_{0}), f_{0} ∈ D(H_{0}), f ∈ D(H).

The assumption (H1) guarantees that the operators GR_{0}(z) : ℌ → 𝔊 and G_{0}(GR_{0}(\overline{z}))^{∗} : 𝔊 → 𝔊 are well-defined and bounded for z ∈ ρ(H_{0}). With slight abuse of notation we put

R_{0}(z)G^{∗} := (GR_{0}(\overline{z}))^{∗}, G_{0}R_{0}(z)G^{∗} := G_{0}(GR_{0}(\overline{z}))^{∗}.

Proposition 3.1. Let H_{0}, H be self-adjoint operators satisfying (3.3) and assume that the operators G_{0}, G satisfy (H1)-(H3). Let z ∈ ρ(H_{0}), then z ∈ ρ(H) iff I + G_{0}R_{0}(z)G^{∗} is boundedly invertible, and in this case

R(z) − R_{0}(z) = −R_{0}(z)G^{∗} (I + G_{0}R_{0}(z)G^{∗})^{−1} G_{0}R_{0}(z).  (3.4)

3.3 Some auxiliary operators

For k ∈ [−\frac{1}{2}, \frac{1}{2}], z ∈ ℂ \ [k^{2}, +∞) we consider in L_{2}(𝕋) the operator

\begin{aligned}D(B(z, k)) & := H^{1}(𝕋), \\ (B(z, k)f)(x_{1}) & := \frac{1}{\sqrt{2π}} ∑_{n∈ℤ} β_{n}(z, k) f̂_{n} e^{inx_{1}},  x_{1} ∈ 𝕋, \end{aligned}  (3.5)

with β_{n}(z, k) defined in (3.2). The operator B(z, k) is invertible for z ∈ ℂ \ [k^{2}, +∞) and its square root is well-defined (it may be considered as the square root of an m-accretive operator). Now, let σ be a periodic function satisfying (1.2) and define operators on L_{2}(𝕋) by

T_{0}^{(σ)}(z, k) := (sgn σ)|σ|^{1/2}B(z, k)^{−1/2}, T^{(σ)}(z, k) := |σ|^{1/2}B(z, k)^{−1/2}.

It follows from Corollary 1.6 that these are compact operators of class Σ_{2}(L_{2}(𝕋)). Finally, for z ∈ ℂ \ [k^{2}, +∞) we consider the integral operator Y (z, k) acting from L_{2}(Π) to L_{2}(𝕋) whose kernel is given by

Y (x_{1}, y; z, k) := \frac{1}{2π} ∑_{n∈ℤ} \frac{1}{\sqrt{β_{n}(z, k)}} e^{in(x_{1}−y_{1})}e^{−β_{n}(z,k) y_{2}}, x_{1} ∈ 𝕋, y ∈ Π.

(3.6)

Writing down the singular value expansion explicitly we find that Y (z, k) is a compact operator of class Σ_{1}(L_{2}(Π), L_{2}(𝕋)).

3.4 The resolvent difference

We are now ready to apply the general results from Subsection 3.2 to our situation. Denote the resolvent of the operator H^{(σ)}(k) by

R^{(σ)}(z, k) := (H^{(σ)}(k) − zI)^{−1} , z ∈ ρ (H^{(σ)}(k)) .

The following statement is of crucial importance.

Proposition 3.2. Let k ∈ [−\frac{1}{2}, \frac{1}{2}] and z ∈ ℂ\[k^{2}, +∞). Then z ∈ ρ (H^{(σ)}(k)) iff the operator I +T_{0}^{(σ)}(z, k)T^{(σ)}(\overline{z}, k)^{∗} is boundedly invertible, and in this case

\begin{matrix}R^{(σ)}(z, k) − R^{(0)}(z, k) = \\ = −Y (\overline{z}, k)^{∗} T^{(σ)}(\overline{z}, k)^{∗} (I + T_{0}^{(σ)}(z, k) T^{(σ)}(\overline{z}, k)^{∗})^{−1} T_{0}^{(σ)}(z, k) Y (z, k). \end{matrix} ; (3.7)

Proof. We want to apply the results of Subsection 3.2 to the case ℌ = L_{2}(Π), 𝔊 = L_{2}(𝕋), H_{0} = H^{(0)}(k), H = H^{(σ)}(k) and

\begin{aligned}D(G_{0}) & = D(G) := H̃^{1}(Π), \\ (G_{0}u)(x_{1}) & := (sgn σ(x_{1}))\sqrt{|σ(x_{1})|} u(x_{1}, 0),  x_{1} ∈ 𝕋, \\ (Gu)(x_{1}) & := \sqrt{|σ(x_{1})|} u(x_{1}, 0),  x_{1} ∈ 𝕋. \end{aligned}

According to Lemma 1.3, the operators G_{0}, G are well-defined and bounded from H̃^{1}(Π) to L_{2}(𝕋). Since (|H_{0}|^{1/2} + I)^{−1} is a bounded operator from ℌ to H̃^{1}(Π), the assumption (H1) in Subsection 3.2 holds. The remaining (3.3), (H2) and (H3) are obvious, so that the statement of Proposition 3.1 holds. Let us determine the products G_{0}R_{0}(z), GR_{0}(z) and G_{0}R_{0}(z)G^{∗}. The explicit form (3.1) of the free resolvent implies that for u ∈ L_{2}(Π)

\begin{aligned}(R_{0}(z)u)(x_{1}, 0) & = \frac{1}{2π} ∑_{n∈ℤ} \frac{1}{β_{n}(z, k)} ∫_{Π} u(y) e^{in(x_{1}−y_{1})} e^{−β_{n}(z,k) y_{2}} dy = \\ & = (B(z, k)^{−1/2} Y (z, k)u) (x_{1}), x_{1} ∈ 𝕋, \end{aligned}

and hence

G_{0}R_{0}(z) = T_{0}^{(σ)}(z, k) Y (z, k), GR_{0}(z) = T^{(σ)}(z, k) Y (z, k).

Moreover, for f ∈ L_{2}(𝕋) we have

(Y (\overline{z}, k)^{∗}f)(x) = \frac{1}{\sqrt{2π}} ∑_{n∈ℤ} \frac{f̂_{n}}{\sqrt{β_{n}(z, k)}} e^{inx_{1}} e^{−β_{n}(z,k) x_{2}}, x ∈ Π,

so that Y (\overline{z}, k)^{∗}f ∈ H̃^{1}(Π) and (Y (\overline{z}, k)^{∗}f)(., 0) = B(z, k)^{−1/2}f. It follows that G_{0}Y (\overline{z}, k)^{∗} = T_{0}^{(σ)}(z, k) and

G_{0}R_{0}(z)G^{∗} = G_{0} (GR_{0}(\overline{z}))^{∗} = T_{0}^{(σ)}(z, k) T^{(σ)}(\overline{z}, k)^{∗}.

This concludes the proof of the Proposition.

As an easy consequence of (3.7) we obtain

Corollary 3.3. Let k ∈ [−\frac{1}{2}, \frac{1}{2}] and z ∈ ρ (H^{(σ)}(k)), then

R^{(σ)}(z, k) − R^{(0)}(z, k) ∈ Σ_{1/3}(L_{2}(Π)).

Proof. We have remarked in Subsection 3.3 that T_{0}^{(σ)}(z, k), T^{(σ)}(z, k) ∈ Σ_{2}(L_{2}(𝕋)) and Y (z, k) ∈ Σ_{1}(L_{2}(Π), L_{2}(𝕋)), so the statement follows from (1.3).

The proof of Theorem 2.1 is now immediate.

Proof of Theorem 2.1. Combine Corollary 3.3 with Proposition 1.1.

3.5 The limiting absorption principle for the unperturbed operator

It remains to prove Theorem 2.2, the absence of singular continuous spectrum. This will be achieved by controlling the behavior of the resolvent R^{(σ)}(z, k) as the spectral parameter z tends to the real axis. We start with the unperturbed case σ ≡ 0.

We introduce the function Λ_{2}(x) := (1 + x_{2}^{2})^{1/2}, x ∈ Π, and note that for k ∈ [−\frac{1}{2}, \frac{1}{2}], z ∈ ℂ \ [k^{2}, +∞) and s > \frac{1}{2} the operator

Λ_{2}^{−s}R^{(0)}(z, k)Λ_{2}^{−s}

belongs to the Hilbert-Schmidt class. The following result is called the limiting absorption principle for the operator H^{(0)}(k).

Proposition 3.4. Let s > \frac{1}{2} and k ∈ [−\frac{1}{2}, \frac{1}{2}]. Then the limits

lim_{ϵ→0+} Λ_{2}^{−s}R^{(0)}(λ ± iϵ, k)Λ_{2}^{−s}, λ ≠ (n + k)^{2}, n ∈ ℤ,

exist in the Hilbert-Schmidt norm and are uniform for λ from compact intervals of ℝ \ {(n + k)^{2} : n ∈ ℤ}.

Proof. Fix λ ∈ ℝ \ {(n + k)^{2} : n ∈ ℤ}, and define

r^{(0)}(x, y; λ ± i0, k) := lim_{ϵ→0+} r^{(0)}(x, y; λ ± iϵ, k), x ≠ y ∈ Π,

as pointwise limit using formula (3.1). We have to prove that

\begin{aligned}_{∫Π ∫Π} & |r^{(0)}(x, y; λ ± iϵ, k) − r^{(0)}(x, y; λ ± i0, k)|^{2} \frac{dx}{(1 + x_{2}^{2})^{s}} \frac{dy}{(1 + y_{2}^{2})^{s}} ⟶ 0 \\ & (ϵ → 0+). \end{aligned} ; (3.8)

We restrict ourselves to the ”+”-case, the other being similar, and for simplicity of notation, we put

c_{n}(ϵ) := β_{n}(λ + iϵ, k), c_{n} := lim_{ϵ→0+} c_{n}(ϵ).

Using Parseval’s identity and the triangle inequality we find

∫Π ∫Π |r^{(0)}(x, y; λ ± iϵ, k) − r^{(0)}(x, y; λ ± i0, k)|^{2} \frac{dx}{(1 + x_{2}^{2})^{s}} \frac{dy}{(1 + y_{2}^{2})^{s}} =

= ∑_{n∈ℤ} ∫_{0}^{∞} \frac{dx_{2}}{(1 + x_{2}^{2})^{s}} ∫_{0}^{∞} \frac{dy_{2}}{(1 + y_{2}^{2})^{s}}

|\frac{e^{−c_{n}(ϵ)(x_{2}+y_{2})} + e^{−c_{n}(ϵ)|x_{2}−y_{2}|}}{2c_{n}(ϵ)} − \frac{e^{−c_{n}(x_{2}+y_{2})} + e^{−c_{n}|x_{2}−y_{2}|}}{2c_{n}} |^{2} ≤

≤ 2 ∑_{n∈ℤ} ∫_{0}^{∞} ∫_{0}^{∞} (t_{n,ϵ}(x_{2} + y_{2}) + t_{n,ϵ}(x_{2} − y_{2})) \frac{dx_{2}}{(1 + x_{2}^{2})^{s}} \frac{dy_{2}}{(1 + y_{2}^{2})^{s}} ,

where

t_{n,ϵ}(a) := |\frac{e^{−c_{n}(ϵ)|a|}}{2c_{n}(ϵ)} − \frac{e^{−c_{n}|a|}}{2c_{n}} |^{2} , a ∈ ℝ.

Documenta Mathematica 8 (2003) 547–565

For (n + k)^{2} < λ it follows from Lebesgue’s theorem that

\begin{matrix}∑_{(n+k)^{2}<λ} & ∫_{0}^{∞} ∫_{0}^{∞} (t_{n,ϵ}(x_{2} + y_{2}) + t_{n,ϵ}(x_{2} − y_{2})) \frac{dx_{2}}{(1 + x_{2}^{2})^{s}} \frac{dy_{2}}{(1 + y_{2}^{2})^{s}} ⟶ 0, \\ (ϵ → 0+). \end{matrix}

Suppose now that (n + k)^{2} > λ. To control the convergence of the t_{n,ϵ} in terms of n we need the elementary estimate

t_{n,ϵ}(a) ≤ ϵ^{2} \frac{C}{|c_{n}|^{6}} , a ∈ ℝ,

with a constant C independent of a, ϵ, λ and n. It follows that

\begin{aligned}∑_{(n+k)^{2}>λ} & ∫_{0}^{∞} ∫_{0}^{∞} (t_{n,ϵ}(x_{2} + y_{2}) + t_{n,ϵ}(x_{2} − y_{2})) \frac{dx_{2}}{(1 + x_{2}^{2})^{s}} \frac{dy_{2}}{(1 + y_{2}^{2})^{s}} ≤ \\ & ≤ 2Cϵ^{2} (∫_{0}^{∞} \frac{dx_{2}}{(1 + x_{2}^{2})^{s}} )^{2} ∑_{n∈ℤ} \frac{1}{|c_{n}|^{6}} . \end{aligned}

The RHS converges to 0 as ϵ → 0+, which completes the proof of (3.8). Finally, we remark that the limit in (3.8) is uniform in λ for λ from a compact intervall not containing any of the points (n + k)^{2}, n ∈ ℤ. This follows from the fact, that c_{n} depends continuously on λ.

3.6 The limiting absorption principle for the perturbed operator

Using the Analytic Fredholm Alternative and the resolvent identity (3.7) we derive the limiting absorption principle for the operator H^{(σ)}(k) from Proposition 3.4.

Lemma 3.5. Let k ∈ [−\frac{1}{2}, \frac{1}{2}], then the operator families

T_{0}^{(σ)}(z, k), T^{(σ)}(z, k), z ∈ ℂ \ [k^{2}, ∞),

can be extended norm-continuously to the cut from above and from below with the exception of the points z = (n+k)^{2}, n ∈ ℤ. Denoting the (upper and lower) boundary values by

T_{0}^{(σ)}(λ ± i0, k),  T^{(σ)}(λ ± i0, k),  λ ≠ (n + k)^{2},

the sets

\begin{matrix}N_{±}(k) := & \{λ ∈ ℝ \ \{(n + k)^{2} : n ∈ ℤ\} : \\ N (I + T_{0}^{(σ)}(λ ± i0, k) T^{(σ)}(λ ∓ i0, k)^{∗}) ≠ \{0\}\} \end{matrix}

are discrete in ℝ \ {(n + k)^{2} : n ∈ ℤ}.

Remark 3.6. One can show that the sets N_{+}(k) and N_{−}(k) coincide. Moreover, in the second part [FrSh] we will prove that the sets N_{±}(k) can accumulate at +∞ only.

Proof. Let

B̃(z, k), z ∈ ℂ \ {(n + k)^{2} + iy : n ∈ ℤ, y ≤ 0} =: D_{+}(k)

be the analytic family of operators given by the same formal expression (3.5) as the operators B(z, k), but where we choose in the definition (3.2) the branch of the square root on ℂ \ {iy : y ≥ 0} which coincides with the canonical branch on the lower halfplane. In particular,

B̃(z, k) = B(z, k),  z ∈ ℂ_{+}.  (3.9)

It follows from Corollary 1.6 that

T̃^{(σ)}(z, k) := |σ|^{1/2}B̃(z, k)^{−1/2}, z ∈ D_{+}(k),

is an analytic family of compact operators. Because of (3.9) it is a bounded analytic (and hence norm-continuous) extension of the family T^{(σ)}(z, k) across the cut from above. We put

T^{(σ)}(λ + i0, k) := T̃^{(σ)}(λ, k), λ ≠ (n + k)^{2}, n ∈ ℤ.

The construction of the operators T^{(σ)}(λ − i0, k) is similar, replacing D_{+}(k) by D_{−}(k) := ℂ \ {(n + k)^{2} + iy : n ∈ ℤ, y ≥ 0}, and the statement about the operators T_{0}^{(σ)}(z, k) follows by multiplying T^{(σ)}(z, k) with sgn σ. Let us prove the statement about the sets N_{±}(k). It follows easily that ∥T̃^{(σ)}(z, k)∥ < 1 if |Im z| is large. Now the Analytic Fredholm Alternative (cf. Theorem VII.1.9 in [K]) applied to the operators T̃_{0}^{(σ)}(z, k) T̃^{(σ)}(\overline{z}, k)^{∗} yields the discreteness of the sets N_{±}(k). This concludes the proof.

Proposition 3.7. Let s > \frac{1}{2} and k ∈ [−\frac{1}{2}, \frac{1}{2}]. Then the limits

lim_{ϵ→0+} Λ_{2}^{−s}R^{(σ)}(λ ± iϵ, k)Λ_{2}^{−s}, λ ∉ {(n + k)^{2} : n ∈ ℤ} ∪ N_{±}(k),

exist in the Hilbert-Schmidt norm and are uniform for λ from compact intervals of ℝ \ ({(n + k)^{2} : n ∈ ℤ} ∪ N_{±}(k)).

Proof. We consider the resolvent identity (3.7). Because of Proposition 3.4 and Lemma 3.5 it suffices to prove that the limits

lim_{ϵ→0+} Y (λ ± iϵ, k)Λ_{2}^{−s}, λ ≠ (n + k)^{2}, n ∈ ℤ,

exist in the Hilbert-Schmidt norm and are uniform for λ from compact intervals of ℝ \ {(n + k)^{2} : n ∈ ℤ}. Considering the integral kernel (3.6) of Y (z, k) one can procede similar to the proof of Proposition 3.4.

As an easy consequence of Proposition 3.7 we obtain now Theorem 2.2.

Proof of Theorem 2.2. Let [a, b] ⊂ ℝ\({(n + k)^{2} : n ∈ ℤ} ∪ N_{±}(k)) be a compact interval. Then for every u from the dense set R(Λ_{2}^{−s}) (with s > \frac{1}{2} arbitrary) we have

sup_{0<ϵ<1} ∫_{a}^{b} |(R^{(σ)}(λ ± iϵ, k)u, u)|^{2} dλ < ∞

by Proposition 3.7. It follows (cf. Proposition 1.5.2 in [Ya]) that the spectrum of H^{(σ)}(k) is purely absolutely continuous on [a, b]. Therefore σ_{sing} (H^{(σ)}(k)) ⊂ {(n+k)^{2} : n ∈ ℤ}∪N_{±}(k). Since the latter set accumulates only at {(n+k)^{2} : n ∈ ℤ} and +∞, we conclude σ_{sc} (H^{(σ)}(k)) = ∅, as claimed.

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Rupert L. Frank Mathematisches Institut LMU München Theresienstraße 39 80333 München, Germany rfrank@mathematik.uni-muenchen.de 566

Documenta Mathematica 8 (2003)