Absolute Continuity of the Spectrum

of a Schrödinger Operator

with a Potential Which is Periodic

in Some Directions and Decays in Others

N. Filonov^{1} and F. Klopp^{2}

Received: January 31, 2004 Revised: March 30, 2004

Communicated by Heinz Siedentop

Abstract. We prove that the spectrum of a Schrödinger operator with a potential which is periodic in certain directions and superexponentially decaying in the others is purely absolutely continuous. Therefore, we reduce the operator using the Bloch-Floquet-Gelfand transform in the periodic variables, and show that, except for at most a set of quasi-momenta of measure zero, the reduced operators satisfies a limiting absorption principle.

2000 Mathematics Subject Classification: 35J10, 35Q40, 81C10

1 Formulation of the result

There are many papers (see, for example, [1, 9]) devoted to the question of the absolute continuity of the spectrum of differential operators with coefficients periodic in the whole space. In the present article, we consider the situation where the coefficients are periodic in some variables and decay very fast (super-exponentially) when the other variables tend to infinity. The corresponding operator describes the scattering of waves on an infinite membrane or filament. Recently, quite a few studies have been devoted to similar problems, for periodic, quasi-periodic or random surface Hamiltonians (see, e.g. [3, 7, 2]).

1N.F.’s research was partially supported by the FNS 2000 “Programme Jeunes Chercheurs”.

2F.K.’s research was partially supported by the program RIAC 160 at Université Paris 13 and by the FNS 2000 “Programme Jeunes Chercheurs”.

Documenta Mathematica 9 (2004) 107–121

Let (x, y) denote the points of the space ℝ^{m+d}. Define Ω = ℝ^{m} × (0, 2π)^{d} and ⟨x⟩ = \sqrt{x^{2} + 1}. For a ∈ ℝ, introduce the spaces

L_{p,a} = {f : e^{a⟨x⟩}f ∈ L_{p}(Ω)}, H_{a}^{2} = {f : e^{a⟨x⟩}f ∈ H^{2}(Ω)},

where 1 ≤ p ≤ ∞ and H^{2}(Ω) is the Sobolev space. Our main result is

Theorem 1.1. Consider in L_{2}(ℝ^{m+d}) the self-adjoint operator

Hu = − div(g∇u) + V u  (1)

and assume that the functions g : ℝ^{m+d} → ℝ and V : ℝ^{m+d} → ℝ satisfy following conditions:

1. ∀l ∈ ℤ^{d}, ∀(x, y) ∈ ℝ^{m+d},

g(x, y + 2πl) = g(x, y), V (x, y + 2πl) = V (x, y);

2. there exists g_{0} > 0 such that (g − g_{0}), Δg, V ∈ L_{∞,a} for any a > 0;

3. there exists c_{0} > 0 such that ∀(x, y) ∈ ℝ^{m+d}, g(x, y) ≥ c_{0}.

Then, the spectrum of H is purely absolutely continuous.

Remark 1.1. Operators with different values of g_{0} differ from one another only by multiplication by a constant; so, without loss of generality, we can and, from now on, do assume that g_{0} = 1.

Remark 1.2. If V ≡ 0, (1) is the acoustic operator. If g ≡ 1, it is the Schrödinger operator with electric potential V.

The basic philosophy of our proof is the following. To prove the absolute continuity of the spectrum for periodic operators (i.e., periodic with respect to a non degenerate lattice in ℝ^{d}), one applies the Floquet-Bloch-Gelfand reduction to the operator and one is left with proving that the Bloch-Floquet-Gelfand eigenvalues must vary with the quasi-momentum i.e., that they cannot be constant on sets of positive measure (see e.g. [9]). If one tries to follow the same line in the case of operators that are only periodic with respect to a sub-lattice, the problem one encounters is that, as the resolvent of the Bloch-Floquet-Gelfand reduction of the operator is not compact, its spectrum may contain continuous components and some Bloch-Floquet-Gelfand eigenvalues may be embedded in these continuous components. The perturbation theory of such embedded eigenvalues (needed to control their behavior in the Bloch quasi-momentum) is more complicated than that of isolated eigenvalues. To obtain a control on these eigenvalues, we use an idea of the theory of resonances (see e.g. [13]): if one analytically dilates Bloch-Floquet-Gelfand reduction of the operator, these embedded eigenvalues become isolated eigenvalues, and thus can be controlled in the usual way.

Let us now briefly sketch our proof. We make the Bloch-Floquet-Gelfand transformation with respect to the periodic variables (see section 3) and get a family of operators H(k) in the cylinder Ω. Then, we consider the corresponding resolvent in suitable weighted spaces. It analytically depends on the quasimomentum k and the spectral (non real) parameter λ. It turns out that we can extend it analytically with respect to λ from the upper half-plane to the lower one (see Theorem 5.1 below) and thus establish the limit absorption principle. This suffices to prove the absolute continuity of the initial operator (see section 7).

Note that an analytic extension of the resolvent of the operator (1) with coefficients g and V which decay in all directions is constructed in the paper [4] (with m = 3, d = 0; see also [10] for g ≡ 1). In the case of a potential decaying in all directions but one (i.e., if d = 1), the analytic extension of the resolvent of the whole operator (1) (not only for the operator H(k) (see section 3)) is investigated in [6] when g ≡ 1. Note also that our approach has shown to be useful in the investigation of the perturbation of free operator in the half-plane by δ-like potential concentrated on a line (see [5]); the wave operators are also constructed there.

In section 2, we establish some auxiliary inequalities. In section 3, we define the Floquet-Gelfand transformation and construct an analytic extension of the resolvent of free operator in the cylinder Ω. In sections 4 and 5, we prove a limiting absorption principle for the initial operator in the cylinder. An auxiliary fact from theory of functions is established in section 6. Finally, the proof of Theorem 1.1 is completed in section 7.

We denote by B_{δ}(k_{0}) a ball in real space

B_{δ}(k_{0}) = {k ∈ ℝ^{d} : |k − k_{0}| < δ}

and by k_{1} the first coordinate of k, k = (k_{1}, k′). We will use the spaces of function in Ω with periodic boundary conditions,

H̃^{2} = \{f ∈ H^{2}(Ω) : f |_{y_{i}=0}= f |_{y_{i}=2π}, \frac{∂f}{∂y_{i}} |_{y_{i}=0}= \frac{∂f}{∂y_{i}} |_{y_{i}=2π}, i = 1, …, d\} ,

H̃_{loc}^{2}= \{f ∈ H_{loc}^{2}(Ω) : f |_{y_{i}=0}= f |_{y_{i}=2π}, \frac{∂f}{∂y_{i}} |_{y_{i}=0}= \frac{∂f}{∂y_{i}} |_{y_{i}=2π}, i = 1, …, d\} .

Finally B(X, Y) is the space of all bounded operators from X to Y, and B(X) = B(X, X), both endowed with their natural topology.

Thanks: the authors are grateful to Prof. P. Kuchment for drawing their attention to the question addressed in the present paper, and to Prof. T. Suslina for useful discussions.

2 Auxiliary estimations

In this section, we assume that the pair (k_{0}, λ_{0}) ∈ ℝ^{d+1} satisfies

(k_{0} + n)^{2} ≠ λ_{0}  ∀n ∈ ℤ^{d}.  (2)

The constants in all the inequalities in this section may depend on (k_{0}, λ_{0}). The set

J = {n ∈ ℤ^{d} : (k_{0} + n)^{2} < λ_{0}}  (3)

is finite. In a neighborhood of (k_{0}, λ_{0}), the partition of ℤ^{d} into J and (ℤ^{d} \ J) is clearly the same. In other words, there exists δ = δ(k_{0}, λ_{0}) > 0 such that

if k ∈ B_{δ}(k_{0}) and λ ∈ B_{δ}(λ_{0}), then (k + n)^{2} < λ ⇔ n ∈ J.  (4)

Choose k̃ ∈ B_{δ}(k_{0}) with k̃_{1} ∉ ℤ and put

k(τ) := (k̃_{1} + iτ, k̃′) ∈ ℂ^{d}, τ ∈ ℝ,

and

M_{1} = M_{1}(k_{0}, λ_{0}) := (B_{δ}(k_{0}) ∪ {k(τ)}_{τ∈ℝ}) × B_{δ}(λ_{0}).  (5)

Lemma 2.1. There exists c > 0 such that, for all ζ ∈ ℝ^{m}, (k, λ) ∈ M_{1}, n ∈ ℤ^{d} \ J and τ ∈ ℝ, we have

|ζ^{2} + (k + n)^{2} − λ| ≥ c,
|ζ^{2} + (k(τ) + n)^{2} − λ| ≥ c|τ|.

Proof. By virtue of (4), there exists c > 0 such that, for n ∈ ℤ \ J,

∀k ∈ B_{δ}(k_{0}), ∀λ ∈ B_{δ}(λ_{0}), (k + n)^{2} − λ > c.

Hence, for ζ ∈ ℝ^{m}, n ∈ ℤ \ J,

∀k ∈ B_{δ}(k_{0}), ∀λ ∈ B_{δ}(λ_{0}), ζ^{2} + (k + n)^{2} − λ > c.

The second inequality is an immediate corollary of our choice of k̃_{1} and the equality

Im(ζ^{2} + (k(τ) + n)^{2} − λ) = 2(k̃_{1} + n_{1})τ.

This completes the proof of Lemma 2.1.

In the remaining part of this section, we assume λ_{0} > 0. In this case, we will need to change the integration path in the Fourier transformation; we now describe the contour deformation. Fix η > \sqrt{λ}_{0} and, let γ be the contour in the complex plane defined as

γ = {−ξ + iη}_{ξ∈[η,∞)} ∪ {α(1 − i)}_{α∈[−η,η]} ∪ {ξ − iη}_{ξ∈[η,∞)}.  (6)

The following two assertions are clear.

Lemma 2.2. If g ∈ L_{2}(γ) and η_{0} > η then the function

h(t) = e^{−η_{0}|t|} ∫_{γ} e^{itz}g(z)dz

belongs to L_{2}(ℝ).

Lemma 2.3. Let Γ denote the open set between real axis and γ (it consists of two connected components). Let g be an analytic function in Γ such that g ∈ C(\overline{Γ}) and |g(z)| ≤ C(1 + | Re z|)^{−2}. Then,

∫_{ℝ} e^{itz}g(z)dz = ∫_{γ} e^{itz}g(z)dz ∀t ∈ ℝ.

Establish an analogue of Lemma 2.1 for n ∈ J and ζ ∈ γ^{m} i.e., ζ = (ζ_{1}, …, ζ_{m}) ∈ ℂ^{m}, ζ_{j} ∈ γ.

Lemma 2.4. Let λ_{0} > 0, η > \sqrt{λ}_{0} and γ be defined by (6). There exists c > 0 such that, for all ζ ∈ γ^{m}, (k, λ) ∈ M_{1}, n ∈ J and τ ∈ ℝ, we have

|ζ^{2} + (k + n)^{2} − λ| ≥ c,
|ζ^{2} + (k(τ) + n)^{2} − λ| ≥ c|τ|.  (7)

Proof. By virtue of (4), there exists c > 0 such that, for n ∈ J,

∀k ∈ B_{δ}(k_{0}), ∀λ ∈ B_{δ}(λ_{0}), (k + n)^{2} − λ < −2c.

Hence, for ζ ∈ γ^{m} such that |ζ| ≤ \sqrt{c}, one has

∀k ∈ B_{δ}(k_{0}), ∀λ ∈ B_{δ}(λ_{0}), Re(ζ^{2} + (k + n)^{2} − λ) < −c.

On the other hand, for ζ ∈ γ^{m} such that |ζ| ≥ \sqrt{c}, one has

∀k ∈ B_{δ}(k_{0}), ∀λ ∈ B_{δ}(λ_{0}), Im(ζ^{2} + (k + n)^{2} − λ) < −c

if one chooses c sufficiently small. Thus, it remains to prove the second inequality. Therefore, we write

ζ^{2} = −2i ∑_{p} α_{p}^{2} + ∑_{q}(ξ_{q} − iη)^{2},

where the indexes p correspond to the coordinates of ζ which are in the middle part of γ (i.e., | Re ζ_{p}| < η) and the indexes q correspond to the extreme parts of γ (i.e., | Re ζ_{q}| ≥ η); it is possible that there are only indexes p or only q. Without loss of generality, we suppose that, for all q, ξ_{q} ≥ 0. Thus,

\begin{aligned}ζ^{2} + (k(τ) + n)^{2} − λ & = ∑_{q}(ξ_{q}^{2} − η^{2}) + (k̃ + n)^{2} − τ^{2} − λ \\ & + 2i (− ∑_{p} α_{p}^{2} − ∑_{q} ξ_{q}η + (k̃_{1} + n_{1})τ) . \end{aligned}

Fix some σ ∈ (η^{−1}\sqrt{λ}_{0}, 1). If ∑_{q} ξ_{q} ≥ σ|τ| then,

|Im(ζ^{2} + (k(τ) + n)^{2} − λ)| ≥ 2 (ση − |k̃_{1} + n_{1}|) |τ| > 2(ση − \sqrt{λ}_{0})|τ|,

as (k̃ + n)^{2} < λ_{0}. If ∑_{q} ξ_{q} ≤ σ|τ| then ∑_{q} ξ_{q}^{2} ≤ σ^{2}τ^{2} and

|Re(ζ^{2} + (k(τ) + n)^{2} − λ)| ≥ τ^{2} + λ − (k̃ + n)^{2} − σ^{2}τ^{2} > (1 − σ^{2})τ^{2}

again by virtue of (4). This completes the proof of Lemma 2.4.

3 The resolvent of free operator in the cylinder

Let us consider the Floquet-Gelfand transformation

(Uf)(k, x, y) = ∑_{l∈ℤ^{d}} e^{i⟨k,y+2πl⟩}f(x, y + 2πl).

It is a unitary operator

U : L_{2}(ℝ^{m+d}) → ∫_{[0,1)^{d}}^{⊕} L_{2}(Ω)dk.

Introduce the family of operators (H(k))_{k∈ℂ^{d}} on the cylinder Ω where for k ∈ ℂ^{d}, Dom H(k) = H̃^{2} and

H(k) = (i∇ − (0, \overline{k}))^{∗} g(x, y) (i∇ − (0, k)) + V (x, y).  (8)

Then, the Schrödinger operator (1) is unitarily equivalent to the direct integral of these operators in Ω:

UHU^{∗} = ∫_{[0,1)^{d}}^{⊕} H(k)dk.

In this section, we investigate the free operator

A(k) = −Δ_{x} + (i∇_{y} − \overline{k})^{∗} (i∇_{y} − k)  (9)

(which corresponds H(k) with g ≡ 1, V ≡ 0). For k ∈ ℝ^{d} and λ ∉ ℝ, its resolvent can be expressed as

((A(k) − λ)^{−1}f) (x, y) = ∑_{n∈ℤ^{d}} ∫_{ℝ^{m}} \frac{e^{iζx+iny}(F f)(ζ, n)dζ}{ζ^{2} + (k + n)^{2} − λ} ,  (10)

where F denotes the Fourier transformation in the cylinder

(F f)(ζ, n) = (2π)^{−m−d} ∫_{Ω} e^{−iζx−iny}f(x, y) dx dy.

Let (k_{0}, λ_{0}) ∈ ℝ^{d+1} satisfy (2) and, J and M_{1} be defined respectively by formulas (3) and (5) in the previous section.

Lemma 3.1. There exists V_{1}, a neighborhood of the set M_{1} in ℂ^{d+1} such that, for (k, λ) ∈ V_{1}, the operator R_{1}(k, λ) given by

(R_{1}(k, λ)f) (x, y) = ∑_{n∈ℤ^{d}\J} ∫_{ℝ^{m}} \frac{e^{iζx+iny}(F f)(ζ, n)dζ}{ζ^{2} + (k + n)^{2} − λ}

is well defined and is bounded from L_{2}(Ω) to H^{2}(Ω). The B(L_{2}(Ω), H^{2}(Ω))- valued function (k, λ) ↦ R_{1}(k, λ) is analytic in V_{1}. For τ ≠ 0, the estimate

∥R_{1}(k(τ), λ)∥_{B(L_{2}(Ω))} ≤ C|τ|^{−1}

holds.

Proof. It immediately follows from Lemma 2.1.

Lemma 3.2. Let λ_{0} > 0, η > \sqrt{λ}_{0}, a > η\sqrt{m} and the contour γ be defined by (6). Then, there exists a neighborhood of the set M_{1}, say V_{2}, such that, for (k, λ) ∈ V_{2}, the operator R_{2}(k, λ) given by

(R_{2}(k, λ)f) (x, y) = ∑_{n∈J} ∫_{γ} ⋯ ∫_{γ} \frac{e^{iζx+iny}(F f)(ζ, n)}{ζ^{2} + (k + n)^{2} − λ} dζ_{1} ⋯ dζ_{m}  (11)

is well defined as a bounded operator from L_{2,a} to H_{−a}^{2}. The B(L_{2,a}, H_{−a}^{2})- valued function (k, λ) ↦ R_{2}(k, λ) is analytic in V_{2}. For τ ≠ 0, the estimate

∥R_{2}(k(τ), λ)∥_{B(L_{2,a}, L_{2,−a})} ≤ C|τ|^{−1}

holds.

Proof. If f ∈ L_{2,a} then the function (F f)(·, n) is square integrable on γ^{m}. By Lemma 2.4, the denominator in (11) never vanishes for (k, λ) ∈ M_{1}; therefore, in some neighborhood of M_{1}. So

|(ζ^{2} + (k + n)^{2} − λ)^{−1}e^{iζx+iny}| ≤ C|e^{iζx}|

where the constant does not depend on ζ ∈ γ^{m} and on x; the same is true for the second derivatives of (ζ^{2} + (k + n)^{2} − λ)^{−1}e^{iζx+iny} with respect to (x, y). Hence, R_{2}(k, λ) ∈ B(L_{2,a}, H_{−a}^{2}) by virtue of Lemma 2.2. Estimation (7) yields the estimation for the norm of R_{2}(k(τ), λ).

Now, we construct an analytic extension of the resolvent of A(k).

Theorem 3.1. Let (k_{0}, λ_{0}) ∈ ℝ^{d+1} satisfy (2) and the set M_{1} be defined in (5). Then, there exists a neighborhood of M_{1}in ℂ^{d+1}, say M_{0}, a real number a and a B(L_{2,a}, H_{−a}^{2})-valued function, say (k, λ) ↦ R_{A}(k, λ), defined and analytic in M_{0}, such that, for (k, λ) ∈ M_{0}, k ∈ ℝ^{d}, Im λ > 0 and f ∈ L_{2,a}, one has

R_{A}(k, λ)f = (A(k) − λ)^{−1}f  (12)

and

∥R_{A}(k(τ), λ)∥_{B(L_{2,a}, L_{2,−a})} ≤ C|τ|^{−1}.  (13)

Proof. If λ_{0} ≤ 0, we can take R_{A} = R_{1} (R_{1} is constructed in Lemma 3.1; here, J = ∅ and a = 0). If λ_{0} > 0 then, we put R_{A} = R_{1}+R_{2}, where R_{1}, R_{2} and a are defined in Lemmas 3.1 and 3.2, and M_{0} is the intersection of V_{1} and V_{2} defined respectively in Lemma 3.1 and Lemma 3.2. If f ∈ L_{2,a} then (F f)(·, n) is an analytic function in the domain \{ζ : | Im ζ| < a\} and is uniformly bounded on \{ζ : | Im ζ| ≤ η\sqrt{m}\}. If ζ ∈ \overline{Γ}^{m} where Γ is the open set between ℝ and γ (see Lemma 2.3), then, Im ζ^{2} ≤ 0; therefore, the integrand in (11) has no poles when Im λ > 0. Hence, the integral in right hand side of (10) for n ∈ J coincides with the corresponding integral in (11) due to Lemma 2.3, and (12) holds. The estimate (13) is a simple corollary of the estimations of Lemmas 3.1 and 3.2.

4 Invertibility of operators of type (I + W R_{A})

Lemma 4.1. Let W ∈ L_{∞,b} for b > 2a > 0. Then, the operator of multiplication by W (we will denote it by the same letter) is

1. bounded as an operator from L_{2,−a} to L_{2,a};

2. compact as an operator from H_{−a}^{2} to L_{2,a}.

Proof. The first assertion is evident. In order to prove the second it is enough to introduce functions

W_{ρ}(x, y) = \begin{cases}W (x, y), & |x| < ρ, \\ 0, & |x| ≥ ρ, \end{cases}

and note that the multiplication by W_{ρ} is a compact operator from H_{−a}^{2} to L_{2,a} and that

∥W − W_{ρ}∥_{B(L_{2,−a},L_{2,a})} → 0

when ρ → ∞.

The next lemma is a well known result from analytic Fredholm theory (see, e.g., [8, 11]).

Lemma 4.2. Let U be a domain in ℂ^{p}, z_{0} ∈ U. Let z ↦ T (z) be an analytic function with values in the set of compact operators in some Hilbert space H. Then, there exists a neighborhood U_{0} of the point z_{0} and an analytic function h : U_{0} → ℂ such that, for z ∈ U_{0},

(I + T (z))^{−1} exists if and only if h(z) = 0.

Now, we can establish the existence of the inverse of (I + W R_{A}).

Theorem 4.1. Let (k_{0}, λ_{0}) satisfy (2), R_{A}(k, λ) and a be defined as in Theorem 3.1. Pick b > 2a, and let (x, y, λ) ↦ W (x, y, λ) be a function which belongs to L_{∞,b} for all λ, and is analytic with respect to λ i.e., λ ↦ W (·, ·, λ) ∈ Hol(ℂ, L_{∞,b}). Then, there exists ε > 0, an open set U ⊂ ℂ^{d+1} such that B_{ε}(k_{0})×B_{ε}(λ_{0}) ⊂ U, and an analytic function h : U → ℂ such that

∀λ ∈ B_{ε}(λ_{0}), ∃k ∈ B_{ε}(k_{0}) such that h(k, λ) ≠ 0,  (14)

and, for any (k, λ) ∈ U, the operator (I + W (λ)R_{A}(k, λ)) is invertible in L_{2,a} if and only if h(k, λ) ≠ 0.

Proof. Due to Theorem 3.1 and Lemma 4.1, the operator W (λ)R_{A}(k(τ), λ) is compact in L_{2,a} and satisfies the inequality

∥W (λ)R_{A}(k(τ), λ)∥_{B(L_{2,a})} ≤ C|τ|^{−1}, ∀λ ∈ B_{ε}(λ_{0}).

Therefore, for |τ| large enough, the operator (I + W (λ)R_{A}(k(τ), λ))^{−1} exists and is bounded on L_{2,a}. The operator-valued function λ ↦ W (λ)R_{A}(k, λ) is analytic in M_{0} (defined in Theorem 3.1). The analytic Fredholm alternative yields that, for each λ ∈ B_{ε}(λ_{0}), one can find k ∈ B_{ε}(k_{0}) such that the operator (I + W (λ)R_{A}(k, λ))^{−1} exists. Now, applying Lemma 4.2 with H = L_{2,a}, z = (k, λ) and T (z) = W R_{A}, completes the proof of Theorem 4.1.

5 The resolvent of the operator H

We can reduce the general case of operator (1) with a “metric” g to the case of “pure” Schrödinger operator due to the following lemma. This identity (for the totally periodic case) is known (see [1]). We include the proof for the convenience of the reader.

Lemma 5.1. Let the operators H(k) and A(k) be defined by (8) and (9) respectively, and let the conditions of Theorem 1.1 be fulfilled with g_{0} = 1. If u ∈ H̃^{2} then,

(H(k) − λ) g^{−1/2}u = g^{1/2} (A(k) + W (λ) − λ) u,

where

W (λ) = \frac{1}{g} (\frac{Δg}{2} − \frac{|∇g|^{2}}{4g} + V + λ(g − 1)) .  (15)

Remark 5.1. If g ≡ 1 then W (λ) ≡ V.

Proof. It is enough to prove the equality

(i∇ − (0, \overline{k}))^{∗} g (i∇ − (0, k)) (g^{−1/2}u) = g^{1/2} (A(k) + \frac{Δg}{2g} − \frac{|∇g|^{2}}{4g^{2}} ) u. (16)

We have

(i∇ − (0, k)) (g^{−1/2}u) = ig^{−1/2}∇u − \frac{i}{2}g^{−3/2}∇gu − (0, k)(g^{−1/2}u).

Therefore, the left hand side of (16) is equal to

(i∇ − (0, \overline{k}))^{∗} (ig^{1/2}∇u − \frac{i}{2}g^{−1/2}∇gu − (0, k)(g^{1/2}u))

= −g^{1/2}Δu + \frac{1}{2} div(g^{−1/2}∇g)u − i⟨k, ∇_{y}(g^{1/2}\overline{u})⟩_{ℂ}

− ig^{1/2}⟨∇_{y}u, \overline{k}⟩_{ℂ} + \frac{i}{2}g^{−1/2}⟨∇_{y}g, \overline{k}⟩_{ℂ}u + k^{2}g^{1/2}u

= g^{1/2} (−Δ_{x}u + (i∇_{y} − \overline{k})^{∗}(i∇_{y} − k)u + \frac{1}{2}g^{−1/2} div(g^{−1/2}∇g)u) .

This completes the proof of Lemma 5.1.

In the following theorem, we describe the meromorphic extension of the resolvent of H(k).

Theorem 5.1. Let the conditions of Theorem 1.1 be fulfilled, the operator H(k) be defined by (8) and (k_{0}, λ_{0}) ∈ ℝ^{d+1} satisfy (2). Then, there exists numbers a ≥ 0, ε > 0, a neighborhood U of (k_{0}, λ_{0}) in ℂ^{d+1} containing the set B_{ε}(k_{0}) × B_{ε}(λ_{0}), a function h ∈ Hol(U) satisfying (14) and an operator-valued function (k, λ) ↦ R_{H}(k, λ) having the following properties:

1. R_{H} is defined on the set {(k, λ) ∈ U : h(k, λ) ≠ 0} and is analytic there;

2. for (k, λ) ∈ U such that h(k, λ) ≠ 0, one has R_{H}(k, λ) ∈ B(L_{2,a}, L_{2,−a});

3. for (k, λ) ∈ U, k ∈ ℝ^{d}, Im λ > 0, f ∈ L_{2,a}, one has

R_{H}(k, λ)f = (H(k) − λ)^{−1}f.  (17)

Remark 5.2. It will be seen from the proof that R_{H}(k, λ) ∈ B(L_{2,a}, H_{−a}^{2}) though we do not need this fact.

Proof. By the assumptions of Theorem 1.1, for any b > 0, ∇g ∈ L_{∞,b}. So, if we define W (λ) by (15), for any b > 0, W (λ) ∈ L_{∞,b}. We can thus apply Theorem 4.1. Let U, h, a and R_{A} be as in this theorem. On the set where h(k, λ) ≠ 0, we put

R_{H}(k, λ) = g^{−1/2}R_{A}(k, λ) (I + W (λ)R_{A}(k, λ))^{−1} g^{−1/2}.

By Theorem 4.1, R_{H}(k, λ) ∈ B(L_{2,a}, H_{−a}^{2}). Let f ∈ L_{2,a}. Then,

(I + W (λ)R_{A}(k, λ))^{−1} g^{−1/2}f ∈ L_{2,a}  (18)

and we can apply Lemma 5.1 to the function

u = R_{A}(k, λ) (I + W (λ)R_{A}(k, λ))^{−1} g^{−1/2}f ∈ H_{−a}^{2},  (19)

so

(H(k) − λ)R_{H}(k, λ)f = g^{1/2} (A(k) + W (λ) − λ) u.  (20)

For real k and non real λ, we have by (12) and (18)

(A(k) − λ)u = (I + W (λ)R_{A}(k, λ))^{−1} g^{−1/2}f,

hence, by (19),

(A(k) + W (λ) − λ) u = g^{−1/2}f,

and, finally, by (20)

(H(k) − λ)R_{H}(k, λ)f = f.  (21)

For Im λ > 0, the operators (H(k) − λ)^{−1} and (A(k) − λ)^{−1} are well defined in L_{2}(Ω). As R_{H}(k, λ)f ∈ L_{2}(Ω), (21) gives R_{H}(k, λ)f = (H(k) − λ)^{−1}f. This completes the proof of Theorem 5.1.

6 One fact from the theory of functions

Lemma 6.1. Let V be an open subset of ℝ^{d}. Let f be a real-analytic function in a box (c, d) × V. Let Λ be a subset of V of measure zero, mes Λ = 0. Then,

mes{k ∈ (c, d) : ∃λ ∈ Λ s.t. f(k, λ) = 0 and ∂_{k}f(k, λ) ≠ 0} = 0.  (22)

Proof. The Implicit Function Theorem implies that, for any point (k_{0}, λ_{0}) such that f(k_{0}, λ_{0}) = 0 ≠ ∂_{k}f(k_{0}, λ_{0}), we can find rational numbers k̃_{0}, r̃_{0} > 0, a vector λ̃_{0} = (λ̃_{0}^{1}, ⋯, λ̃_{0}^{d}) with rational coordinates, and a cube Cr̃_{0}(k̃_{0}, λ̃_{0}) where

(k_{0}, λ_{0}) ∈ Cr̃_{0}(k̃_{0}, λ̃_{0}) = (k̃_{0} − r̃_{0}, k̃_{0} + r̃_{0}) × Cr̃_{0}(λ̃_{0}) ⊂ (c, d) × V
Cr̃_{0}(λ̃_{0}) = (λ̃_{0}^{1} − r̃_{0}, λ̃_{0}^{1} + r̃_{0}) × ⋯ × (λ̃_{0}^{d} − r̃_{0}, λ̃_{0}^{d} + r̃_{0})

and a real analytic function θ : Cr̃_{0}(λ̃_{0}) → (k̃_{0} − r̃_{0}, k̃_{0} + r̃_{0}) such that

1. θ(λ_{0}) = k_{0};

2. f(k, λ) = 0 ⇔ θ(λ) = k if (k, λ) ∈ Cr̃_{0}(k̃_{0}, λ̃_{0}).

Therefore,

mes{k : ∃λ ∈ Λ s.t. (k, λ) ∈ Cr̃_{0}(k̃_{0}, λ̃_{0}) and f(k, λ) = 0}
≤ mes θ(Λ ∩ Cr̃_{0}(λ̃_{0})) = 0.

The set {(k, λ) : f(k, λ) = 0 and ∂_{k}f(k, λ) ≠ 0} can be covered by a countable number of cubes Cr̃(k̃, λ̃) constructed as above, say (Cr̃_{i}(k̃_{i}, λ̃_{i}))_{i∈ℕ}; hence, the measure of the set under consideration in (22) is also equal to zero as

{k ∈ (c, d) : ∃λ ∈ Λ s.t. f(k, λ) = 0 and ∂_{k}f(k, λ) ≠ 0}

⊂ ⋃_{i∈ℕ} mes{k : ∃λ ∈ Λ s.t. (k, λ) ∈ Cr̃_{i}(k̃_{i}, λ̃_{i}) and f(k, λ) = 0}.

This completes the proof of Lemma 6.1.

Lemma 6.1 has a multidimensional analogue.

Lemma 6.2. Let U be an open subset of ℝ^{d}, and V be an open subset of ℝ^{d′}. Let f be a real-analytic function on the set U × V, and pick Λ ⊂ V such that mes Λ = 0. For k ∈ U, we write k = (k_{1}, k′) where k_{1} is real and k′ ∈ ℝ^{d−1}. Then,

mes{k ∈ U : ∃λ ∈ Λ s.t. f(k, λ) = 0 and ∂_{k_{1}}f(k, λ) ≠ 0} = 0.  (23)

Proof. Cover U with countably many open sets of the form (a, b) × Ũ i.e.,

U = ⋃_{i∈ℕ}(a_{i}, b_{i}) × Ũ_{i}.

For i ∈ ℕ, one has

{k ∈ (a_{i}, b_{i}) × Ũ_{i} : ∃λ ∈ Λ s.t. f(k, λ) = 0 and ∂_{k_{1}}f(k, λ) ≠ 0}
⊂ {k_{1} ∈ (a_{i}, b_{i}) : ∃λ ∈ Λ s.t. f(k_{1}, k′, λ) = 0 and
∂_{k_{1}}f(k_{1}, k′, λ) ≠ 0} × Ũ_{i}.

(24)

By Lemma 6.1, the set in the right hand side of equation (24) has measure 0 (as Ũ_{i} × Λ has measure zero in ℝ^{d+d′−1}). As

{k ∈ U : ∃λ ∈ Λ s.t. f(k, λ) = 0 and ∂_{k_{1}}f(k, λ) ≠ 0}

= ⋃_{i∈ℕ}{k ∈ (a_{i}, b_{i}) × Ũ_{i} : ∃λ ∈ Λ s.t. f(k, λ) = 0 and ∂_{k_{1}}f(k, λ) ≠ 0},

(23) holds, which completes the proof of Lemma 6.2.

Finally, we prove

Theorem 6.1. Let U be a region in ℝ^{d}, Λ be a subset of an interval (a, b) such that mes Λ = 0. Let h be a real-analytic function defined on the set U × (a, b) and suppose that

∀λ ∈ Λ ∃k ∈ U such that h(k, λ) ≠ 0.  (25)

Then, mes{k ∈ U : ∃λ ∈ Λ s.t. h(k, λ) = 0} = 0.

Proof. For any k ∈ U and λ ∈ Λ, by assumption (25), there exists a multi-index α ∈ ℤ_{+}^{d} such that ∂_{k}^{α}h(k, λ) ≠ 0. Therefore,

{k ∈ U : h(k, λ) = 0 for some λ ∈ Λ}
⊂ ⋃_{j=1}^{d} ⋃_{α∈ℤ_{+}^{d}} {k ∈ U : ∂_{k}^{α}h(k, λ) = 0, ∂_{k_{j}}∂_{k}^{α}h(k, λ) ≠ 0 for some λ ∈ Λ} .

Reference to Lemma 6.2 then completes the proof of Theorem 6.1.

7 The proof of Theorem 1.1

The following lemma is well known (see for example [12]).

Lemma 7.1. Fix b > 0. Let B be a self-adjoint operator in L_{2}(Ω). Suppose that R_{B} is an analytic function defined in a complex neighborhood of an interval [α, β] except at a finite number of points {μ_{1}, …, μ_{N}}, that the values of R_{B} are in B(L_{2,b}, L_{2,−b}) and that

R_{B}(λ)φ = (B − λ)^{−1}φ if Im λ > 0, φ ∈ L_{2,b}.

Then, the spectrum of B in the set [α, β] \ {μ_{1}, …, μ_{N}} is absolutely continuous. If Λ ⊂ [α, β], mes Λ = 0 and μ_{j} ∉ Λ, j = 1, …, N, then E_{B}(Λ) = 0, where E_{B} is the spectral projector of B.

Proof of Theorem 1.1. By Theorem 5.1, the set of all points (k, λ) ∈ ℝ^{d+1} satisfying (2) can be represented as the following union

{(k, λ) ∈ ℝ^{d+1} s.t. (2) be satisfied} = ⋃_{j=1}^{∞} B_{ε_{j}}(k_{j}) × B_{ε_{j}}(λ_{j}),  (26)

where, for every j, there exists

• a number a_{j} ≥ 0,

• an analytic scalar function h_{j} defined in a complex neighborhood of \overline{B_{ε_{j}}(k_{j}) × B_{ε_{j}}(λ_{j})} with the property

∀λ ∈ B_{ε_{j}}(λ_{j}) ∃k ∈ B_{ε_{j}}(k_{j}) such that h_{j}(k, λ) ≠ 0,

• an analytic B(L_{2,a_{j}}, L_{2,−a_{j}})-valued function R_{H}^{(j)} defined on the set where h_{j}(k, λ) ≠ 0 and satisfying (17).

Now, pick Λ ⊂ ℝ such that mes Λ = 0. Set

K_{0} = {k ∈ [0, 1]^{d} : (k + n)^{2} = λ for some n ∈ ℤ^{d}, λ ∈ Λ},
K_{1} = {k ∈ [0, 1]^{d} : h_{j}(k, λ) = 0 for some j ∈ ℕ, λ ∈ Λ}.

Thanks to Theorem 6.1, we know

mes K_{0} = mes K_{1} = 0.  (27)

For k ∉ K_{0}, denote

Λ_{j}(k) = {λ ∈ Λ : (k, λ) ∈ B_{ε_{j}}(k_{j}) × B_{ε_{j}}(λ_{j})}.

It is clear that Λ_{j}(k) ⊂ (λ_{j} − ε_{j}, λ_{j} + ε_{j}), mes Λ_{j}(k) = 0, and, by (26),

Λ = ⋃_{j=1}^{∞} Λ_{j}(k) ∀k ∉ K_{0}.  (28)

If k ∉ (K_{0}∪K_{1}) and Λ_{j}(k) ≠ ∅ then h_{j}(k, λ) ≠ 0 for λ ∈ Λ_{j}(k) and λ ↦ h_{j}(k, λ) has at most a finite number of zeros in [λ_{j} − ε_{j}, λ_{j} + ε_{j}]. So we can apply Lemma 7.1; therefore,

E_{H(k)}(Λ_{j}(k)) = 0 ∀j.

This and (28) implies that

E_{H(k)}(Λ) = 0.

Finally, one computes

E_{H}(Λ) = ∫_{[0,1]^{d}} E_{H(k)}(Λ) dk = ∫_{[0,1]^{d}\K_{0}\K_{1}} E_{H(k)}(Λ) dk = 0

by virtue of (27). So, we proved that the spectral resolution of H vanishes on any set of Lebesgue measure 0, which means, by definition, that the spectrum of the operator H is purely absolutely continuous.

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N. Filonov

Department of

Mathematical Physics St Petersburg

State University 198504 St Petersburg-

Petrodvorets Russia filonov@mph.phys.spbu.ru F. Klopp LAGA, Institut Galilée Université Paris-Nord F-93430 Villetaneuse France klopp@math.univ-paris13.fr 122

Documenta Mathematica 9 (2004)