ON DISCRETE SPECTRUM IN THE GAPS OF A TWO-DIMENSIONAL ...

ON DISCRETE SPECTRUM IN THE GAPS OF A TWO-DIMENSIONAL PERIODIC ELLIPTIC OPERATOR PERTURBED BY A DECAYING POTENTIAL

T. A. Suslina

Abstract.

We study the discrete spectrum in the gaps of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. We are interested in the asymptotics (for large coupling constant) of the number of eigenvalues that have been “born” (or have “died”) at the edges of the gap. The high-energy (Weyl) asymptotics and the threshold asymptotics are distinguished. At the right edge of the gap the competition between the Weyl contribution and the threshold contribution to the asymptotics is possible.

§ 0. Introduction 1. Let A be a selfadjoint elliptic periodic second order operator in L2 (Rd ), d ≥ 2, given by the expression A = −div g(x)∇ + p(x), and let V be an operator of multiplication by a function V (x) ≥ 0 decaying at infinity. Let the interval (λ− , λ+ ) be a gap in the spectrum of A. We put A± (α) = A ∓ αV (x), α > 0. By N+ (α, λ+ ) we denote the number of eigenvalues of the operator A+ (t) that have been ”born” at the point λ+ as the coupling constant t has been growing from 0 to α. The function N− (α, λ− ) is defined similarly for the operator A− . We are interested in the asymptotics of these functions as α → ∞ (in the large coupling constant limit). The corresponding asymptotics may be rather diverse, depending both on A and V . They were studied in a number of papers. First of all, we mention [B3, B4, BL] and especially the survey [B2] and the references therein. Another approach to the class of problems under discussion was proposed in [Iv]. The asymptotic behavior of N± (α, λ± ) depends on dimension d and on the character of decay of V . The situation for d ≥ 3 differs substantially from that for d = 2. If d ≥ 3 and V ∈ Ld/2 (Rd ), the function N+ (α, λ+ ) has the Weyl asymptotics Z N+ (α, λ+ ) ∼ (2π)−d ωd αd/2 V d/2 (det g)−1/2 dx, α → ∞; (0.1) here ωd is the volume of the unit ball in Rd . If V 6∈ Ld/2 (Rd ), then the estimate N+ (α, λ+ ) = O(αd/2 ) is violated, and N+ (α, λ+ ) can have an arbitrary order of growth greater than d/2. Essentially, the asymptotics (0.1) has a “high-energy” origin, while the behavior of N+ (α, λ+ ) with V 6∈ Ld/2 (Rd ) is determined by the “threshold” effect near the edge of the gap of the unperturbed operator A. (See discussion in [B2, §2].) 1991 Mathematics Subject Classification. Primary 35P20. Key words and phrases. Periodic operator, perturbation, discrete spectrum, threshold effect. Supported by RFBR (grant no. 02-01-00798) 1

2

T. A. SUSLINA

2. If d = 2, the situation is much more complicated. Let V ∈ L1 (R2 ). Already for A = −∆ in the case of the semi-infinite gap (−∞, 0), this condition is insufficient for (0.1). Due to the threshold effects, N+ (α, 0) may have an arbitrary order of growth greater than d/2. Moreover, it may happen that N+ (α, 0) = O(α), but the asymptotics is not of Weyl type. In this case the asymptotic coefficient is the sum of the Weyl term and the “threshold” term. A “special channel” is responsible for the threshold effect. We mean the problem on the semi-axis that is obtained by restriction of −∆ − αV to the subspace of functions depending only on |x|. At the same time, the potential V is averaged over the polar angle. These effects were investigated in [BL] in detail. At the level of estimates this channel was discovered before in [S]. In [BLSu], the same effects were studied in the case where A is a periodic elliptic operator of the form A = −div g(x)∇ + p(x). Adding an appropriate constant to p allows us to assume that the lower edge of the spectrum is the point λ = 0. In [BLSu], the negative discrete spectrum of the operator A − αV , i. e., the case of the semi-bounded gap (−∞, 0) was studied. The description of the special channel was given in terms of the Floquet-Bloch decomposition for the unperturbed operator A. The answer involves the so-called tensor of effective masses at the edge of the spectrum and a positive periodic solution ϕ of the equation Aϕ = 0. The function ϕ can be eliminated from the answer under an additional “regularity” condition imposed on V . The present paper is a continuation of [BLSu], but now we study the case of an internal gap in the spectrum of A. For this, we need to change the technique of investigation substantially. 3. For the study of the functions N± (α, λ± ) in an internal gap of A, we impose certain restrictions on the structure of the edges of the gap (see Condition 1.3(±) below). For the lower edge of the spectrum λ = 0 this condition is fulfilled automatically. The answers are given in terms of the model operators, which are simpler than A± (α). The model operators involve the tensors of effective masses at the edges of the gap and the corresponding eigenfunctions. As well as for the semiinfinite gap, it is possible to eliminate the eigenfunctions from the answer under an additional “regularity” condition imposed on V . The main results are formulated in Theorems 2.2(±), 2.5(±). On the right (but not on the left) edge of the gap a competition between the Weyl contribution and the threshold contribution to the asymptotics is possible. 4. Acknowledgments. The author is grateful to M. Sh. Birman for useful discussions and constant attention to work. 5. Notation. In what follows, Q2 is an open unit square in R2 . The symbol h·, ·i stands for the standard inner product in Cm ; 1 is the unit (2 × 2)-matrix. Any integral without indication of the integration domain is over R2 . Further, ∇ = grad, ∇∗ = −div. We denote by H s , s ≥ 0, the Sobolev classes. Many statements and formulas contain the double indices “±”. Unless otherwise explicitly stated, the upper and the lower versions should be read independently. § 1. Setting of the problem. Preliminaries 1. Differential operators. The unperturbed operator A is formally given by the expression Au = ∇∗ g∇u + pu. Here g is a (2 × 2)-matrix-valued function, p is a

ON DISCRETE SPECTRUM IN THE GAPS

3

real-valued function. We assume that g = g > 0, p = p, g + g −1 ∈ L∞ (R2 ), p ∈ L∞ (R2 ),

)

g(x + n) = g(x), p(x + n) = p(x), x ∈ R2 , n ∈ Z2 .

(1.1)

There is no loss of generality in assuming that the lattice of periods is Z2 . The precise definition of A as a selfadjoint operator in the Hilbert space L2 (R2 ) is given via the closed lower semi-bounded quadratic form Z a[u, u] = (hg∇u, ∇ui + p|u|2 ) dx, u ∈ H 1 (R2 ). (1.2) Adding an appropriate constant to p allows usRto assume that inf spec A = 0. Under this condition, in H 1 (R2 ) the form a[u, u] + γ |u|2 dx, γ > 0, determines a metric equivalent to the standard one. A perturbation is introduced as the operator of multiplication by a function V (x) such that V (x) ≥ 0, x ∈ R2 . We impose the following condition on V (cf., e. g., [BL]). Condition 1.1. For some σ > 1,  1/σ  Z X   |V |σ dx +   |x|≤1

k≥1

Z

1/σ  |V |σ |x|2(σ−1) dx

< ∞.

ek−1 ≤|x|≤ek

We mention at once that (1.3) implies that  1/σ X Z  ((V ))σ := V σ dx < ∞, σ > 1,  n∈Z2

(1.3)

(1.4)

Q2 +n

and, moreover, V ∈ L1 (R2 ). Consider the quadratic form Z v[u, u] = V |u|2 dx. Under condition (1.4) (and, moreover, under condition (1.3)), this form is compact in H 1 (R2 ). Consequently, the form a± (α)[u, u] := a[u, u] ∓ αv[u, u], u ∈ H 1 (R2 ), α > 0, is lower semi-bounded and closed in L2 (R2 ). The form a± (α) generates a selfadjoint operator A± (α) in L2 (R2 ). Formally, the operator A± (α) corresponds to the differential expression A± (α)u = ∇∗ g∇u + pu ∓ αV u. The spectrum of A± (α) in the spectral gaps of A is discrete. First, we recall the result for the semi-infinite gap. Let N+ (α, λ; A, V ), α > 0, λ ≤ 0, denote the number of eigenvalues of the operator A+ (α), lying to the left of the point λ. For the Weyl asymptotic coefficient we introduce the notation Z (1.5) J(V, g) := (4π)−1 V (det g)−1/2 dx.

4

T. A. SUSLINA

Proposition 1.2. Under condition (1.4), we have N+ (α, λ; A, V ) ≤ Cα((V ))σ , C = C(g, p, σ, λ), σ > 1, λ < 0, lim α−1 N+ (α, λ; A, V ) = J(V, g), λ < 0.

α→∞

(1.6)

Comments on Proposition 1.2 and necessary references can be found in [BLSu]. For λ = 0 the Weyl asymptotics (1.6) may fail to occur even under condition (1.3) because of spectral “threshold” effects. These phenomena were studied in the paper [BL] for the operator −∆−αV and in [BLSu] in the general case of a periodic operator A. Below we shall impose one more condition on V (see Condition 2.1(q)), which ensures that N+ (α, 0; A, V ) = O(αq ), α → ∞, q ≥ 1. In the present paper the discrete spectrum of the operators A± (α) in internal gaps of A is studied. 2. The Floquet decomposition. As usual, for the periodic operators we employ e 1 (Q2 ) be the subspace formed by the functions in the Floquet-Bloch theory. Let H 1 2 2 1 H (Q ) whose Z -periodic extensions belong to the class Hloc (R2 ). Next, we denote 1 2 2 e (Q ), ξ ∈ R , the subspace of functions of the form u(x) = eihx,ξi v(x), v ∈ by H ξ e 1 (Q2 ). Consider the following family of quadratic forms in L2 (Q2 ): H Z e ξ1 (Q2 ), ξ ∈ R2 . aξ [u, u] = (hg∇u, ∇ui + p|u|2 ) dx, u ∈ H (1.7) Q2

The selfadjoint operator in L2 (Q2 ) generated by the form (1.7) is denoted by A(ξ). The operator A(ξ) corresponds to the expression A with (ξ)-quasi-periodic boundary conditions. Usually it is sufficient to consider ξ ∈ T2 = R2 /(2πZ)2 . The parameter ξ is called the quasi-momentum. All operators A(ξ) have discrete spectrum. Let Es (ξ), s ∈ N, be consecutive eigenvalues (counted with multiplicity) of the operator A(ξ); and let ψs (x, ξ) be the corresponding eigenfunctions normalized in L2 (Q2 ). The functions Es are continuous and (2πZ)2 -periodic. The spectrum of A coincides with the union of the intervals (bands) that are the ranges of the functions Es . The eigenfunctions ψs admit representation of the e 1 (Q2 ). The functions ψs , ϕs are H¨older form ψs (x, ξ) = eihx,ξi ϕs (x, ξ), ϕs (·, ξ) ∈ H continuous in x. We consider the integral operators Z −1 (Ψs u)(ξ) = (2π) ψs (x, ξ)u(x) dx, s ∈ N. The mappings Ψs : L2 (R2 ) → L2 (T2 ) are partially isometric and surjective. The operators Ψ∗s Ψs , P s ∈ N, are orthoprojections in L2 (R2 ). They are orthogonal to one another and s∈N Ψ∗s Ψs = I. Denoting by [Es ] the operator of multiplication P by the function Es (ξ) in L2 (T2 ), we have A = s∈N Ψ∗s [Es ]Ψs . 3. A gap. The spectrum of A may have gaps other than the semi-infinite gap (−∞, 0). Let Λ = (λ− , λ+ ) be a gap. Clearly, λ+ = min2 El (ξ),

(1.8+)

λ− = max2 El−1 (ξ),

(1.8−)

ξ∈T

ξ∈T

for some l ∈ N. We impose certain restrictions on the “structure” of the edges of the gap. Let us formulate condition for λ+ .

ON DISCRETE SPECTRUM IN THE GAPS

5

Condition 1.3(+). a) minξ∈T2 El+1 (ξ) > λ+ ; b) The minimum (1.8+) is at(+) tained only at finitely many points ξj ∈ T2 , j = 1, . . . , m+ , all of which are non-degenerate minimum points for El (·). Remark 1.4. For the semi-infinite gap (−∞, 0) Condition 1.3(+) at λ+ = 0 is (+) fulfilled automatically with l = 1, m+ = 1, ξ1 = 0. This fact was used in [BLSu]. By Condition 1.3(+), λ+ is a simple eigenvalue of the operator A(ξ) with ξ = (+) ξj , j = 1, . . . , m+ . Then, for some (sufficiently small) δ > 0 the eigenvalue (+)

El (ξ), |ξ − ξj | ≤ δ, is simple. This implies the real analyticity of El (ξ) in these (+)

neighborhoods of the points ξj , j = 1, . . . , m+ . Then Condition 1.3(+), b) means that (+)

(+)

(+)

(+)

El (ξ)−λ+ = bj (ξ−ξj )+O(|ξ−ξj |3 ), |ξ−ξj | ≤ δ, j = 1, . . . , m+ , (1.9+) (+)

where bj is a positive definite quadratic form. Let us formulate condition on λ− . Condition 1.3(−). a) maxξ∈T2 El−2 (ξ) < λ− ; b) The maximum in (1.8−) is (−) attained only at finitely many points ξj ∈ T2 , j = 1, . . . , m− , all of which are non-degenerate maximum points for El−1 (·). Similarly to (1.9+), Condition 1.3(−), b) means that (−)

(−)

(−)

(−)

λ− − El−1 (ξ) = bj (ξ − ξj ) + O(|ξ − ξj |3 ), |ξ − ξj | ≤ δ, j = 1, . . . , m− , (1.9−) (−) where bj is a positive definite quadratic form. (±)

Remark 1.5. We agree that the points ξj (±)

semi-open cube: ξj (±)

of points ξj

∈ T2 are represented as points of the

∈ [−π, π)2 , j = 1, . . . , m± . Accordingly, small neighborhoods (±)

∈ T2 are understood as R2 -neighborhoods of points ξj (±)

∈ [−π, π)2 .

(±)

The form bj (ξ − ξj ) can be written as (±)

(±)

(±)

(±)

(±)

(±)

bj (ξ − ξj ) = hbj (ξ − ξj ), ξ − ξj i = |βj (±)

βj (±)

where bj

(±)

(ξ − ξj )|2 ,

(±)

= (bj )1/2 , j = 1, . . . , m± ,

(1.10±)

(±)

is a constant positive definite matrix. (The matrix (bj )−1 determines (±)

the tensor of effective masses for the point ξj .) We put E+ := El , E− := El−1 , ψ (+) := ψl , ϕ(+) := ϕl , ψ (−) := ψl−1 , (−) ϕ := ϕl−1 . The functions ψ (±) , ϕ(±) can be chosen as real-analytic H 1 (Q2 )(±) valued functions of ξ for |ξ − ξj | ≤ δ, j = 1, . . . , m± . The functions ψ (±) , ϕ(±) are H¨older continuous in x. We introduce the notation (±)

ψj

(±)

(±)

(±)

(x) := ψ (±) (x, ξj ), ϕj (x) := ϕ(±) (x, ξj ), j = 1, . . . , m± .

(1.11±)

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T. A. SUSLINA

4. Let λ ∈ (λ− , λ+ ) = Λ. We denote by N± (α, λ; A, V ), α > 0, λ ∈ Λ,

(1.12)

the number of eigenvalues of A± (t) that have crossed λ as the coupling constant t has been growing from 0 to α. The following statement was proved in [B1] (see also [B2, Theorem 3.2]). Proposition 1.6. Under condition (1.4), we have lim α−1 N+ (α, λ; A, V ) = J(V, g), λ ∈ Λ,

α→∞

(1.13)

lim α−1 N− (α, λ; A, V ) = 0, λ ∈ Λ.

α→∞

Thus, if the “observation point” λ lies inside the gap, then N+ has the Weyl asymptotics (1.13). For λ = λ+ the asymptotics (1.13) may fail to occur even under condition (1.3). Below we impose an additional condition (Condition 2.1(q)) on V that ensures the finiteness of the following limits: N± (α, λ+ ; A, V ) = N± (α, λ− ; A, V ) =

lim

N± (α, λ; A, V ),

(1.14+)

lim

N± (α, λ; A, V ),

(1.14−)

λ→λ+ −0

λ→λ− +0

We are interested in the behavior of functions (1.14) as α → ∞. 5. On compact operators. Let H be a separable Hilbert space. The space of continuous linear operators is denoted by R, and that of compact operators by S∞ . Let T ∈ S∞ , and let sk (T ) be the singular numbers of T , i. e., the consecutive eigenvalues (counted with multiplicity) of the operator (T ∗ T )1/2 . We denote n(s, T ) := card {k : sk (T ) > s}, s > 0. For T = T ∗ , we put 2T± = |T | ± T and n± (s, T ) := n(s, T± ), s > 0. Clearly, (+) n+ (·, T ) is the counting function for the sequence {λk (T )} of positive eigenvalues (−) (−) of T . For the sequence {λk (T )} a similar role is played by n− (·, T ) (here λk (T ) = (+) λk (−T )). We have n(s, T ) = n+ (s, T ) + n− (s, T ), s > 0. We denote by Σq , 0 < q < ∞, the space (ideal) of compact operators distinguished by the condition T

q q

:= sup sq n(s, T ) < ∞, q > 0. s>0

The space Σq is complete in the quasi-norm · Σq we consider the functionals

q

and non-separable. On the space

∆q (T ) := lim sup sq n(s, T ), s→0

δq (T ) := lim inf sq n(s, T ). s→0

(1.15)

ON DISCRETE SPECTRUM IN THE GAPS

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The condition ∆q (T ) = 0 determines the separable subspace Σ0q in Σq . For T = T ∗ ∈ Σq , we put (±) ∆(±) q (T ) := ∆q (T± ), δq (T ) := δq (T± ).

(1.16)

Below Dq stands for any of the functionals (1.15), (1.16). (+) (−) If T ∗ = T ∈ S∞ (H), then the numbers λk (T ) (the numbers (−λk (T ))) coincide with the consecutive positive maxima (the negative minima) of the ratio of quadratic forms ± (1.17) (T u, u)H kuk2H , u ∈ H. Passage from T to the ratio (1.17) facilitates using variational arguments. Therefore, we shall use the simpler notation n± (s, (1.17)) in place of n± (s, T ), (1.17) q in place of T q , Dq (1.17) in place of Dq (T ), etc. 6. An auxiliary problem on the semi-axis. Let f = f ∈ L1,loc (R+ ). For some R ≥ 1, consider the ratio of quadratic forms ,Z∞ Z∞ f (r)|z(r)|2 r dr |z 0 (r)|2 r dr , z(R) = 0, R ≥ 1. (1.18) R

R

Here z runs through all functions absolutely continuous on R+ and such that the integral in the denominator is finite. On f we impose the following “implicit” condition: for some q ≥ 1, (1.18) q < ∞, q ≥ 1.

(1.19)q

This condition is fulfilled (or not fulfilled) simultaneously for all R ≥ 1. Moreover, under condition (1.19)q all six functionals Dq (1.18) do not depend on R ≥ 1. We can give an elementary sufficient condition for (1.19)q (see [BL], and also [BS], [BLSu]). This condition becomes necessary for the non-negative f . Namely, we put ζ(f ) := {ζn (f )}, n ∈ Z+ , Z1

Ze

n

|f (et )|e2t dt, ζn (f ) :=

ζ0 (f ) := 0

t|f (et )|e2t dt, n ∈ N ; en−1

kζ(f )kqq,∞ := sup sq card{n : ζn (f ) > s}, q ≥ 1, s>0

∆q (ζ(f )) := lim sup sq card{n : ζn (f ) > s}, q ≥ 1, s→0

δq (ζ(f )) := lim inf sq card{n : ζn (f ) > s}, q ≥ 1. s→0

Proposition 1.7. a) Assume that kζ(f )kq,∞ < ∞, q ≥ 1.

(1.20)q

Then (1.19)q is true, and ∆q (1.18) ≤ C(q) ∆q (ζ(f )). b) Assume that f (r) ≥ 0, r ≥ R0 , for some R0 ≥ 1. Then (1.19)q implies condition (1.20)q and also the inequalities ∂q (1.18) ≥ c(q) ∂q (ζ(f )), ∂ = ∆, δ.

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T. A. SUSLINA

§ 2. Formulation of the main results 1. Our goal is to study the asymptotics of the functions N± (α, λ+ ; A, V ), N± (α, λ− ; A, V ) (see (1.14)) as α → ∞. We introduce the following quantities: −q ∆(±) N± (α, λτ ; A, V ), τ = ±, q ≥ 1, q (λτ ; A, V ) := lim sup α

(2.1τ )

δq(±) (λτ ; A, V ) := lim inf α−q N± (α, λτ ; A, V ), τ = ±, q ≥ 1.

(2.2τ )

α→∞

α→∞

For a function F (x), x ∈ R2 , we put Fβ (x) = F (βx); here β is a positive matrix. Let (r, θ) be the polar coordinates of a point x ∈ R2 ; we write F (x) = F (r, θ). We put Zπ −1 hF i(r) = (2π) F (r, θ) dθ. −π

Along with Condition 1.1, we impose the following condition on V . Condition 2.1(q). For some q ≥ 1, (1.18) q < ∞ with f = hV i. Since V ≥ 0, Condition 2.1(q) is equivalent (see Proposition 1.7) to the following relation: kζ(hV i)kq,∞ < ∞. Examples (for any q ≥ 1) demonstrating the compatibility of Conditions 1.1 and 2.1(q) can be found in [BL] and [BLSu, §8]. For ϕ ∈ L∞ (R2 ), we introduce the notation fβ,ϕ := h(|ϕ|2 V )β i. Let (+) ∆(+) q (V, β, ϕ), δq (V, β, ϕ), q ≥ 1,

(2.3)

(+)

(+)

denote the functionals ∆q (1.18), δq (1.18) for f = fβ,ϕ . We mention that Condition 2.1(q) with f = hV i is equivalent to the same condition with f = fβ,ϕ . Also, we note that the functionals (2.3) coincide for potentials asymptotically close as |x| → ∞ (see Proposition 2.2 from [BLSu]). 2. In [BLSu] it was shown that, if V satisfies Conditions 1.1 and 2.1(q), then N+ (α, 0; A, V ) = O(αq ), α → ∞,

(2.4)

and the corresponding asymptotic formulas were established. Earlier the same was established in [BL] in the case where A = −∆. Below we formulate two theorems (Theorems 2.2(τ ) and 2.4(τ )) about the asymptotics of N± (α, λτ ; A, V ), τ = ±. In Theorems 2.2(±) the answers are formulated in terms of the corresponding model Schr¨ odinger operators with, generally speaking, matrix-valued potentials. In Theorems 2.4(±) the answers are formulated in terms of the auxiliary problem on the semi-axis, but V is subject to the additional restriction. (±) The description of the model operators involves the quadratic forms bj (see (±)

(1.9)) and the corresponding eigenfunctions ψj (see (1.11)). In the Hilbert space H± = L2 (R2 ; Cm± ), we consider the following diagonal second order elliptic operator with constant coefficients: (±)

B± (D) = diag (b1 (D), . . . , b(±) m± (D)), D = −i∇.

(2.5±)

ON DISCRETE SPECTRUM IN THE GAPS

9

The expression (2.5±) generates a positive selfadjoint operator B± in H± . Now, we introduce the matrix-row and the matrix-column (±)

Π± (x) := {ψj

m

(±)

± (x)}j=1 , Π∗± (x) := col {ψj

m

± (x)}j=1 .

We denote W (x) = (V (x))1/2 and define the non-negative matrix potential U± (x) := (±) (W (x)Π± (x))∗ W (x)Π± (x). The functions ψj (x) are bounded; therefore, the potential U± (x) admits a point-wise estimate in terms of V (x). Now, we introduce the model operators B± (α) := B± − α U± (x), α > 0.

(2.6±)

By N+ (α, λ; B± , U± ), α > 0, λ ≤ 0, we denote the number of eigenvalues of the operator B± (α) lying to the left of the point λ. The estimate (2.4) for A = −∆ is carried over to the operator (2.6): N+ (α, 0; B± , U± ) = O(αq ), α → ∞. Consider the ratio of (finite-dimensional) forms hU± (x)c, ci , c ∈ Cm± ; x ∈ R2 , η ∈ R2 . (2.7±) hB± (η)c, ci For µ > 0, by n(±) (µ; x, η) we denote the number of eigenvalues of the ratio (2.7±) that are greater than µ. For λ < 0, the function N+ (α, λ; B± , U± ) has the Weyl asymptotics Z Z e ± , U± ) := (2π)−2 lim α−1 N+ (α, λ; B± , U± ) = J(B n(±) (1; x, η) dx dη, λ < 0.

α→∞

(2.8±) We introduce the notation ∆q (B± , U± ) := lim sup α−q N+ (α, 0; B± , U± ), q ≥ 1,

(2.9±)

δq (B± , U± ) := lim inf α−q N+ (α, 0; B± , U± ), q ≥ 1,

(2.10±)

e 1 (B± , U± ) := ∆1 (B± , U± ) − J(B e ± , U± ), ∆ e ± , U± ). δe1 (B± , U± ) := δ1 (B± , U± ) − J(B

(2.11±)

α→∞

α→∞

(2.12±)

Theorem 2.2(+). Let the operator A be generated by the form (1.2) under conditions (1.1). Let (λ− , λ+ ) be a gap in the spectrum of A. Suppose that Condition 1.3(+) is satisfied. Suppose that the potential V ≥ 0 satisfies Conditions 1.1 and 2.1(q). Then the following is true for the quantities (2.1+), (2.2+): (a) If q = 1, then ∂1 (λ+ ; A, V ) = J(V, g) + ∂e1 (B+ , U+ ), (+)

∂ = ∆, δ,

(2.13)

(−)

∆1 (λ+ ; A, V ) = 0. (2.14) e e 1 (B+ , U+ ), δ1 (B+ , U+ ) are defined in (2.11+), Here J(V, g) is as in (1.5), and ∆ (2.12+). For the validity of the Weyl asymptotics (+)

(+)

∆1 (λ+ ; A, V ) = δ1 (λ+ ; A, V ) = J(V, g) (+)

(2.15)

(+)

it suffices that ∆1 (V, 1, 1) = 0. Here ∆1 (V, 1, 1) is defined in (2.3). (b) If q > 1, then (2.14) is fulfilled and ∂q(+) (λ+ ; A, V ) = ∂q (B+ , U+ ),

∂ = ∆, δ.

Here ∆q (B+ , U+ ), δq (B+ , U+ ) are defined in (2.9+), (2.10+).

(2.16)

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T. A. SUSLINA

Theorem 2.2(−). Let the operator A be generated by the form (1.2) under conditions (1.1). Let (λ− , λ+ ) be a gap in the spectrum of A. Suppose that Condition 1.3(−) is satisfied. Suppose that the potential V ≥ 0 satisfies Conditions 1.1 and 2.1(q). Then the following is true for the quantities (2.1−), (2.2−): (a) If q = 1, then (+)

(+)

∆1 (λ− ; A, V ) = δ1 (λ− ; A, V ) = J(V, g),

(2.17)

∂1 (λ− ; A, V ) = ∂e1 (B− , U− ), ∂ = ∆, δ.

(2.18)

(−)

e 1 (B− , U− ), δe1 (B− , U− ) are defined in (2.11−), Here J(V, g) is as in (1.5), and ∆ (2.12−). (b) If q > 1, then (+) ∆1 (λ− ; A, V ) ≤ J(V, g), (2.19) ∂q(−) (λ− ; A, V ) = ∂q (B− , U− ), ∂ = ∆, δ.

(2.20)

Here ∆q (B− , U− ), δq (B− , U− ) are defined in (2.9−), (2.10−). (±)

3. The model operator (2.6±) involves the forms bj ces

(±) βj

(see (1.10)), and also the eigenfunctions

the dependence on

(±) βj

or, equivalently, the matri-

(±) ψj .

It is impossible to avoid

in formulas (2.13), (2.16), (2.18), (2.20). Concerning the (±)

(more unpleasant) dependence on the functions ψj , the things are different. It is possible to eliminate these functions from the asymptotic formulas under some supplementary conditions of “regular” behavior of the perturbation V . We proceed to the formulation of Theorem 2.4 that solves this problem. In addition to Conditions 1.1, 2.1(q), we impose the following condition on V . Condition 2.3. There exists a function S = S satisfying Condition 1.1 (with V replaced by S) and such that V (x) = S(x)(1 + o(1)) as |x| → ∞. Suppose that the Fourier-image ΦS of S satisfies the following condition: for some κ > 1, ΦS ∈ H κ (R2 \ Bε ), ∀ε > 0, where Bε = {ξ ∈ R2 : |ξ| ≤ ε}. Theorem 2.4(+). Let the operator A be generated by the form (1.2) under conditions (1.1). Let (λ− , λ+ ) be a gap in the spectrum of A. Suppose that Condition 1.3(+) is satisfied. Suppose that the potential V ≥ 0 satisfies Conditions 1.1, 2.1(q), 2.3. Then the following is true for the quantities (2.1+), (2.2+): (a) If q = 1, then (2.14) is satisfied and (+) ∂1 (λ+ ; A, V

) = J(V, g) +

m+ X

(+)

(+)

∂1 (V, βj

, 1),

∂ = ∆, δ.

(2.21)

j=1 (+)

(+)

Here J(V, g) is as in (1.5), and ∂1 (V, βj , 1), ∂ = ∆, δ, are defined in accordance with (2.3). (b) If q > 1, then (2.14) is satisfied and ∂q(+) (λ+ ; A, V

)=

m+ X j=1

(+)

∂q(+) (V, βj

, 1),

∂ = ∆, δ.

(2.22)

ON DISCRETE SPECTRUM IN THE GAPS

11

Theorem 2.4(−). Let the operator A be generated by the form (1.2) under conditions (1.1). Let (λ− , λ+ ) be a gap in the spectrum of A. Suppose that Condition 1.3(−) is satisfied. Suppose that the potential V ≥ 0 satisfies Conditions 1.1, 2.1(q), 2.3. Then the following is true for the quantities (2.1−), (2.2−): (a) If q = 1, then (2.17) is satisfied and (−)

∂1 (λ− ; A, V ) =

m− X

(+)

(−)

, 1), ∂ = ∆, δ.

(2.23)

(−)

, 1), ∂ = ∆, δ.

(2.24)

∂1 (V, βj

j=1

(b) If q > 1, then (2.19) is satisfied and ∂q(−) (λ− ; A, V ) =

m− X

∂q(+) (V, βj

j=1

§ 3. Sketch of the proof 1. Reduction to compact operators. We denote X(λ) := W (A − λI)−1 W, λ ∈ Λ.

(3.1)

The functions (1.12) are related to the counting functions of the spectrum of the operator X(λ): N± (α, λ; A, V ) = n± (t, X(λ)), tα = 1, λ ∈ Λ.

(3.2)

In (3.2) we cannot pass to the limit as λ → λ± , because the operators (3.1) do not have limits. Therefore, we need an appropriate regularization. Proposition 3.1(±). Suppose that near λ± the operator X(λ) is represented in the form X(λ) = Γ± (λ) + Y± (λ), (3.3±) where Γ± (λ) = Γ± (λ)∗ , the limit (with respect to the operator norm) (u)- lim Γ± (λ) =: Γ± ∈ Σq , q ≥ 1, λ→λ±

exists, and (uniformly in λ) rank Y± (λ) ≤ r± < ∞. Then ∂q(+) (λ± ; A, V ) = ∂q(+) (Γ± ), ∂ = ∆, δ, ∂q(−) (λ± ; A, V ) = ∂q(−) (Γ± ), ∂ = ∆, δ.

(3.4±)

2. For N > 0, let ζN (x) be the characteristic function of the disc {|x| ≤ N }, and fN := ζeN W , VN := ζN V , let ζeN (x) := 1 − ζN (x). We denote WN := ζN W , W VeN := ζeN V . The operator X(λ) is represented in the form X(λ) = LN (λ) + KN (λ) + 2Re MN (λ), fN (A − λI)−1 W fN , MN (λ) := where LN (λ) := WN (A − λI)−1 WN , KN (λ) := W −1 fN (A − λI) WN . We are going to regularize these operators separately and exW amine the contribution of each of them to the limit quantities (2.1), (2.2). Below we suppose that Conditions 1.1, 2.1(q) and 1.3(±) are satisfied. 3. The operator LN (λ) is “responsible” for the Weyl contribution to the asymptotics.

12

T. A. SUSLINA

Proposition 3.2(±). We have (±) b (±) (λ), LN (λ) = LN (λ) + L N (±)

b (λ) ≤ 2m± , the limit where rank L N (±)

(±)

(Σ1 )- lim LN (λ) =: LN (λ± ) ∈ Σ1 λ→λ±

exists, and (+)

(±)

(+)

(±)

∆1 (LN (λ± )) = δ1 (LN (λ± )) = J(VN , g), (−)

(3.5±)

(±)

∆1 (LN (λ± )) = 0. (±)

(±)

We have LN (λ± ) = ζN LN (λ± )ζN . 4. The operator KN (λ) is responsible for the threshold contribution to the (±) (±) (±) asymptotics. We put Ej := {ξ : |βj (ξ − ξj )| ≤ δ}, j = 1, . . . , m± . Choose a number δ > 0 so small that E± (ξ) is a simple eigenvalue of the operator A(ξ) (±) (±) (±) (±) for ξ ∈ Ej , j = 1, . . . , m± , and Ej ∩ Ek = ∅ for j 6= k. Let χj denote the (±)

characteristic function of the ellipse Ej We introduce the projections X

(±)

:=

m± X

.

(±) Ψ∗± [χj ]Ψ± , Xe(±) := I − X (±) ,

j=1

which commute with A. Here Ψ+ := Ψl , Ψ− := Ψl−1 . The operator KN (λ) is represented in the form (±)

(±)

KN (λ) = QN (λ) + PN (λ),

(3.6±)

(±) fN (A − λI)−1 X (±) W fN , P (±) (λ) := W fN (A − λI)−1 Xe(±) W fN . where QN (λ) := W N (±) The operator PN (λ) in the limit N → ∞ gives no contribution to the asymptotics.

Proposition 3.3(±). The limit (±)

(±)

(Σ1 )- lim PN (λ) =: PN (λ± ) ∈ Σ1 λ→λ±

exists, and (+) (±) (+) (±) ∆1 (PN (λ± )) = δ1 (PN (λ± )) = J(VeN , g), (−)

(±)

∆1 (PN (λ± )) = 0. (±)

The operator QN (λ) can be written as follows: (±)

QN (λ) =

m± X j,k=1

(±)

(±)

(TjN (λ)Ψ± )sign (A − λI)(TkN (λ)Ψ± )∗ ,

(3.7±)

ON DISCRETE SPECTRUM IN THE GAPS (±)

13

(±)

fN Ψ∗ [χ |E± − λ|−1/2 ], j = 1, . . . , m± . TjN (λ) := W ± j (±)

The integral operator TjN (λ) has the following kernel: (±)

fN (x)χ (ξ) |E± (ξ) − λ| (2π)−1 W j

−1/2 ihx,ξi

e

ϕ(±) (x, ξ).

(±)

Under regularization, the operator TjN (λ) can be replaced by the integral operator (±) Tb (λ) with the following simpler kernel: jN

¯−1/2 ¯ (±) fN (x)χ(±) (ξ) ¯¯±b(±) (ξ − ξ(±) ) + λ± − λ¯¯ eihx,ξi ϕj (x). (2π)−1 W j j j (±)

(±)

are defined in (1.9), and ϕj are introduced in (1.11). It is (±) elementary to reduce the operator TbjN (λ) to the model integral operator studied in [BLSu, §3]. The results of [BLSu] imply the following statement.

Recall that bj

Proposition 3.4(±). The following representations are valid: (±) (±) (±) TbjN (λ) = TjN (λ) + YjN (λ), (±)

where rank YjN (λ) = 1 and the limit (±)

(±)

(u)- lim TjN (λ) =: TjN (λ± ) ∈ Σ2q λ→λ±

exists. As a result, we obtain the following statement for the operator (3.7±). Proposition 3.5(±). We have (±) (±) b (±) (λ), QN (λ) = QN (λ) + Q N

b (±) (λ) ≤ 2m± , the limit where rank Q N (±)

(±)

(u)- lim QN (λ) =: QN (λ± ) ∈ Σq λ→λ±

(±)

(±)

exists, and QN (λ± ) = RN (±) RN

:= ±

(mod Σ0q ), where m± X

(±)

(±)

(TjN (λ± )Ψ± )(TkN (λ± )Ψ± )∗ .

(3.8±)

j,k=1 (+)

(+)

(+)

(−)

(−)

(−)

The quantities ∂q (RN ) =: ∂q (∗) and ∂q (RN ) =: ∂q (∗), ∂ = ∆, δ, do not depend on N . Relation (3.6±) and Propositions 3.3(±), 3.5(±) imply the following statement.

14

T. A. SUSLINA

Proposition 3.6(±). We have (±) b (±) (λ), KN (λ) = KN (λ) + K N

b (±) (λ) ≤ 2m± and the limit where rank K N (±)

(±)

(u)- lim KN (λ) =: KN (λ± ) ∈ Σq λ→λ±

exists. Moreover, (±)

lim ∂q(±) (KN (λ± )) = ∂q(±) (∗),

(3.9±)

N →∞

(∓)

(±)

lim ∆1 (KN (λ± )) = 0.

N →∞ (±)

(±)

We have KN (λ± ) = ζeN KN (λ± )ζeN . 5. The operator MN (λ) in the limit N → ∞ gives no contribution to the asymptotics. Proposition 3.7(±). We have (±) c(±) (λ), MN (λ) = MN (λ) + M N (±)

c (λ) ≤ 2m± , the limit where rank M N (±)

(±)

(u)- lim MN (λ) =: MN (λ± ) ∈ Σq λ→λ±

exists, and (±)

lim ∆q (MN (λ± )) = 0.

(3.10±)

N →∞

6. Now everything is prepared for applying the general method of Subsection 1. The operator X(λ) is represented in the form (3.3±) with the operators (±)

(±)

(±)

(±)

ΓN (λ) := LN (λ) + KN (λ) + 2Re MN (λ), (±)

(±)

(±)

(±)

b (λ) + K b (λ) + 2Re M c (λ) YN (λ) := L N N N (±)

in the role of Γ± (λ), Y± (λ). Herewith, rank YN (λ) ≤ 6m± and the limit (±)

(±)

(±)

(±)

(±)

(u)- lim ΓN (λ) =: ΓN (λ± ) = LN (λ± ) + KN (λ± ) + 2Re MN (λ± ) (3.11±) λ→λ±

(±)

exists. We write relations (3.4±) for ΓN (λ± ): (±)

∂q(+) (λ± ; A, V ) = ∂q(+) (ΓN (λ± )), ∂ = ∆, δ, (±)

∂q(−) (λ± ; A, V ) = ∂q(−) (ΓN (λ± )), ∂ = ∆, δ.

(3.12±)

ON DISCRETE SPECTRUM IN THE GAPS

15

Since the left-hand sides in (3.12±) are independent of N , so are the right-hand sides. We pass to the limit in (3.12±) as N → ∞. Relations (3.5±), (3.9±), (3.10±), (3.11±) and (3.12±) with q > 1 imply that ∂q(±) (λ± ; A, V ) = ∂q(±) (∗), ∂ = ∆, δ,

(3.13±)

∆(∓) q (λ± ; A, V ) = 0.

(3.14±)

If q = 1, the same relations yield the following equalities: (+)

(+)

∂1 (λ+ ; A, V ) = J(V, g) + ∂1 (∗), ∂ = ∆, δ, (+)

(3.15+)

(+)

∆1 (λ− ; A, V ) = δ1 (λ− ; A, V ) = J(V, g),

(3.15−)

(−)

∆1 (λ+ ; A, V ) = 0, (−)

(3.16+)

(−)

∂1 (λ− ; A, V ) = ∂1 (∗), ∂ = ∆, δ.

(3.16−)

Now it is easy to complete the proof of Theorem 2.2, applying a similar regularization for the model operators and comparing the results with formulas (3.13±)– (3.16±). 7. Scheme of the proof of Theorem 2.4. Below ∂ = ∆, δ. The additional (±) Condition 2.3 on V allows us to calculate ∂q (∗) (cf. (3.8±)). Under this condition, we have (±) (±) (TjN (λ± )Ψ± )(TkN (λ± )Ψ± )∗ ∈ Σ0q , j 6= k. (±)

Then ∂q (∗) is equal to the sum of the corresponding quantities for the operators (±) (±) (TjN (λ± )Ψ± )(TjN (λ± )Ψ± )∗ . In [BLSu, §3], it was shown that these functionals are related to the auxiliary problem on the semi-axis. They coincide with (±) (±) (±) ∂q (V, βj , ϕj ). Condition 2.3 allows us to eliminate (cf. [BLSu, §7]) the (±)

functions ϕj (±)

(±)

∂q (V, βj

(±)

(±)

from the asymptotic coefficients. We have ∂q (V, βj

(±)

, ϕj ) =

, 1). This leads to the asymptotic formulas (2.21)–(2.24). References

[B1]

[B2]

[B3]

[B4]

[BL]

Birman M. Sh., Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations, Adv. Soviet Math. 7 (1991), Amer. Math. Soc., Providence, RI, 57-73. Birman M.Sh., Discrete spectrum of the periodic Schr¨ odinger operator perturbed by a decaying potential, Algebra i Analiz 8 (1996), no. 1, 3-20; English transl., St. Petersburg Math. J. 8 (1997), no. 1, 1–14. Birman M.Sh., The discrete spectrum in gaps of the perturbed Schr¨ odinger operator. I. Regular perturbations, Boundary value problems, Schr¨ odinger operators, deformation quantization, Math. Top. 8 (1995), Akademie Verlag, Berlin, 334-352. Birman M.Sh., Discrete spectrum in the gaps of a perturbed periodic Schr¨ odinger operator. II. Nonregular perturbations, Algebra i Analiz 9 (1997), no. 6, 62-89; English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1073–1095. Birman M.Sh., Laptev A., The negative discrete spectrum of a two-dimensional Schr¨ odinger operator, Comm. Pure Appl. Math. 49 (1996), 967-997.

16

T. A. SUSLINA

[BLSu] Birman M.Sh., Laptev A., Suslina T. A., Discrete spectrum of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. I. Semibounded gap, Algebra i Analiz 12 (2000), no. 4, 36-78; English transl., St. Petersburg Math. J. 12 (2001), no. 4, 535–567. [BS] Birman M.Sh., Solomyak M.Z., Estimates for the number of negative eigenvalues of the Schr¨ odinger operator and its generalizations, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations, Adv. Soviet Math. 7 (1991), Amer. Math. Soc., Providence, RI, 1-55. [Iv] Ivrii V., Accurate spectral asymptotics for periodic operators, Journ´ ees “Equations aux derivees partielles” (1999), Saint-Jean-de-Monts, 1-11. [S] Solomyak M. Z., Piecewise-polynomial approximation of functions from H l ((0, 1)d ), 2l = d, and applications to the spectral theory of the Schr¨ odinger operator, Israel J. Math. 86 (1994), 253–275.

St. Petersburg State University, Physics Department, Ul'yanovskaya 1, Petrodvorets, 198904, St. Petersburg, Russia E-mail address: [email protected]