In the semiclassical regime, we obtain a lower bound for the counting function of resonances corresponding to the perturbed periodic Schrödinger operator Ph=-Δ+Vx+W(hx). Here V is a periodic potential, W a decreasing perturbation and h a small positive constant.

1. Introduction

The quantum dynamics of a Bloch electron in a crystal subject to external electric field, which varies slowly on the scale of the crystal lattice, is governed by the Schrödinger equationP(h)=-Δ+V(x)+W(hx).
Here V is periodic with respect to the crystal lattice Γ⊂ℝn, and it models the electric potential generated by the lattice of atoms in the crystal. The potential W is a decreasing perturbation and h a small positive constant.

There has been a growing interest in the rigorous study of the spectral properties of Bloch electrons in the presence of slowly varying external perturbations (see [1–11]).

Since the work of Peierls [10] and Slater [11], it is well known that, if h is sufficiently small, then solutions of P(h) are governed by the “semiclassical” Hamiltonian
H(y,η)=λ(η+A(y))+V(y).
Here λ(k) is one of the “band functions” describing the Floquet spectrum of the unperturbed HamiltonianP0=-Δx+V(x).
One argues that for suitable wave packets, which are spread over many lattice spacings, the main effect of a periodic potential on the electron dynamics consists in changing the dispersion relation from the free kinetic energy Efree(k)=|k|2 to the modified kinetic energy λ(k) given by the Bloch band.

The problem of resonances has been examined in [12] for the one-dimensional case and in [13] for the general case. In particular, a similar reduction to (1.2) for resonances has been obtained in [13].

This paper continues our previous works [13, 14] on the resonances and the eigenvalues counting function for P(h). In [14], Dimassi and Zerzeri obtained a local trace formula for resonances. As a consequence, they obtained an upper bound for the number of resonances of P(h) in any h-independent complex neighborhood of some energy E. The purpose of this paper is to give a lower bound for the number of resonances of P(h).

In the case where V=0, it is known that, for 0<E in the analytic singular support (from now on sing suppa for short) of the distribution dρ0*μ, then the operator P(h)=-Δ+W(hx) has at least CΩh-n resonances in any h-independent complex neighborhood Ω of E (see, e.g., [15]). Here μ(t)=∫{x∈Rn;W(x)>t}dx,ρ0(t)=(2π)-nvol(B(0,1))(max(t,0))n/2.

Using the explicit formula of ρ0 we see that the analytic singular support of the distributions μ and dρ0*μ coincide.

In the case where V≠0 the situation is different. Following Theorem 1.6 in [14] and Lemma 2.1 of the next section, we have to change ρ0 by
ρ(λ):=1(2π)n∑j≥1∫{k∈E*;λj(k)≤λ}dk,
which is the integrated density of states corresponding to the nonperturbed Hamiltonian P0 (see Section 2).

If λj(k) is a simple eigenvalue near some point e0, then λj(k) is a smooth function, and if e0=λj(k) is a critical value, we expect in general that e0 will belong to the analytic singular support of ρ(λ). In particular, we expect that near every point e∈e0+sing suppa(μ) there exists at least Ch-n, C>0, resonances.

Multiple eigenvalues (λj(k0)=λj+1(k0)=e0) can also give rise to singularities of ρ(λ) and then lead to the existence of resonances near e0+sing suppa(μ).

The purpose of this paper is to describe all these situations. Some results of this paper are announced without proofs in [16].

The paper is organized as follows: in the next section, we introduce some notations and state some technical lemmas. In Section 3 we give an upper bound for resonances near singularities of the density of states measure ρ generated by a band crossing. In Section 4 we give an upper bound for resonances near the edge of bands.

2. Preliminaries

Let Γ=⊕i=1nℤai be the lattice generated by the basis a1,a2,…,an, ai∈ℝn. The dual lattice Γ* is defined as the lattice generated by the dual basis {a1*,a2*,…,an*} determined by ai·aj*=2πδij, i,j=1,2,…,n. Let E be a fundamental domain for Γ, and let E* be a fundamental domain for Γ*. If we identify opposite edges of E (resp., E*), then it becomes a flat torus denoted by 𝕋=ℝn/Γ (resp., 𝕋*=ℝn/Γ*).

Let V be a real valued potential, C∞ and Γ-periodic. For k in ℝn, we defineP0(k)=(Dx+k)2+V(x)
as an unbounded operator on L2(𝕋) with domain H2(𝕋). The Hamiltonian P0(k) is semibounded and self-adjoint. Since the resolvent of (Dx+k)2 is compact, the resolvent of P0(k) is also compact, and therefore P0(k) has a complete set of (normalized) eigenfunctions Φn(·,k)∈H2(𝕋*), n∈ℕ, called Bloch functions. The corresponding eigenvalues accumulate at infinity, and we enumerate them according to their multiplicities:λ1(k)≤λ2(k)≤⋯.
Since e-ixγ*H0(k)eixγ*=H0(γ*+k), the band function λn(k) is periodic with respect to Γ*. The function λn(k) is called a band function, and the closed intervals Λn:=λn(𝕋*) are called bands.

Standard perturbation theory shows that λn(k) is a continuous function of k and is real analytic in a neighborhood of any k such thatλn-1(k)<λn(k)<λn+1(k).
We fix λ in the spectrum of the unperturbed operator P0. We make the following hypothesis on the spectrum of the unperturbed Schrödinger operator.

For all k0 with λi(k0)=λ, the eigenvalue λi(k0) is simple and dkλi(k0)≠0.

Now, let us recall some well-known facts about the density of states associated with P0. The density of states measure ρ is defined as follows:ρ(λ):=1(2π)n∑j≥1∫{k∈E*;λj(k)≤λ}dk,
where E* is a fundamental domain of ℝn/Γ*. Since the spectrum of P0 is absolutely continuous, the measure ρ is absolutely continuous with respect to the Lebesgue measure dλ. Thus, the density of states of P0, ∂ρ/∂λ is locally integrable.

We now consider the perturbed periodic Schrödinger operator:P(h):=P0+W(hx),
where W∈C∞(ℝn;ℝ). We assume that there exist positive constants a and C such that W extends analytically to Γ(a):={z∈ℂn;|ℑ(z)|≤a〈ℜ(z)〉} and|W(z)|≤C〈z〉-ñ,uniformly onz∈Γ(a),ñ>n,
where 〈z〉=(1+|z|2)1/2. Here ℜ(z),ℑ(z) denote, respectively, the real part and the imaginary part of z.

This assumption allows us to define the resonances of P(h) by the spectral deformation method (see [17]). We follow essentially the presentation of [13].

Let v∈C∞(ℝn;ℝn) be Γ*-periodic. For t∈ℝ, we introduce the spectral deformation family 𝒰t defined by for all u∈𝒮:Utu(r):=Fh-1{(Jt1/2(Fhu)(vt(k)))}(r),∀x∈Rn,
where vt(k)=k-tv(k) and Jt(k) its Jacobian. Here ℱh is the semiclassical Fourier transform:[Fhu](ξ):=∫Rne-(i/h)xξu(x)dx,∀u∈S(Rn).
Consider, for t∈ℝ, the family of unitarily equivalent operatorsP1(t,h):=UtP1(h)Ut-1.
It was established in [13, Proposition 2.8] that P1(t,h) extends to an analytic type-𝒜 family of operators on D(t0):={t∈ℂ;|t|<t0} with domain H2(ℝn). Moreover, under the assumptions (H1) and (2.6), there exists a neighborhood Ω̃ of z0 and a small positive constant η such that, for t∈D(t0) with ℑt>0, the spectrum of P1(t,h) in Ωt:={z∈Ω̃;ℑz>-ηℑt} consists of discrete eigenvalues of finite multiplicities that lie in the lower half plane (see [13, formula (4.9)]). These eigenvalues are t-independent under small variations of ℑt>0 and are called resonances. We will denote the set of resonances by Res(P(h)).

For f∈C0∞(ℝ), we set
〈μ,f〉=∫[f(W(x))-f(0)]dx,〈ω,f〉=1(2π)n∑j≥1∫E*∫Rxn[f(W(x)+λj(k))-f(λj(k))]dkdx,
For E>0, letν+(E):=∫{x∈Rn;W(x)≥E}dx.
Similarly, for E<0, we setν-(E):=∫{x∈Rn;W(x)≤E}dx.
Clearly, ν+(E) (resp., ν-(E)) is a decreasing function of E (resp., an increasing function of E) andμ|R±=-ddEν±(E).

Lemma 2.1.

The distributions ω and μ are real valued of order ≤1. Moreover, in𝒟′(ℝ), one has

ω=dρ*μ.

Proof.

Applying Taylor’s formula to the right-hand side of (2.10), we obtain
|〈μ,f〉|≤sup|f′|∫|W(x)|dx,
which together with (2.6) imply that μ is a distribution of order ≤1, with
suppμ⊂[infW(x),supW(x)].
Consequently, dρ*μ is well defined in 𝒟′(ℝ) and for all f∈C0∞(R), we have
〈dρ*μ,f〉=〈dρ(t),〈μ,f(⋅+t)〉〉=-〈ρ(t),∫[f′(W(x)+t)-f′(t)]dx〉=-1(2π)n∑j∫E*∫λj(k)∞∫Rxn[f′(W(x)+t)-f′(t)]dxdtdk=1(2π)n∑j∫E*∫Rxn[f(W(x)+λj(k))-f(λj(k))]dxdk=〈ω,f〉.
This ends the proof of the lemma.

Let Ω be an open-bounded set in ℝn, and let Ω̃ be a complex neighborhood of Ω. Let x→φ(x) be analytic on Ω̃ and real valued for all x in Ω. Let us introduce the real functionI(e):=∫{x∈Ω;φ(x)≤e}dx.
For e∈φ(Ω), we setΣ(e):={x∈Ω;φ(x)=e}.

Lemma 2.2.

Let e0∈φ(Ω), and let Σ(e), I(e) be as above. One assumes that

∇φ(x)≠0 for all x∈Σ(e0),

∂Ω∩Σ(e0)=∅.

Then the function
I(e):=∫{x∈Ω;φ(x)≤e}dx
is analytic near e0.

Proof.

Let ϵ be a small positive constant such that ∇φ(x)≠0 when x∈Σϵ(e0):=φ-1(]e0-ϵ,e0+ϵ[). Without any loss of generality we may assume that ∂x1φ≠0 for all x∈Σϵ(e0). By the change of variable H:x→(φ(x),x2,…,xn) we have
∫{x∈Σϵ(e0):;φ(x)≤e}dx=∫{x∈H(Σϵ(e0):);x1≤e}Jac(H-1(x))dx.
Clearly the right-hand side of the above equality is analytic. Combining this with the fact that ∫{x∈Ω∖Σϵ(e0):;φ(x)≤e}dx is constant for e near e0 we get the lemma.

Lemma 2.3.

If φ has a nondegenerate extremum at x0 with φ(x0)=e0 and if ∇φ(x)≠0 for all x∈Σe0∖{x0}, then
I(e)=f(e-e0)+H(±(e-e0))g(±(e-e0)),
where f and g are analytic near zero and
g(t)~t→0vol(Sn-1)ndetφ′′(x0)2n/2tn.
Here H(t) is the Heaviside function and +(-) corresponds to a minimum (maximum, resp.).

Proof.

Here we only give a sketch of the proof. For the details we refer to [18]. Without any loss of generality, we only consider the case of minimum. By Morse lemma there exist a neighborhood U of x0, ϵ>0 and a local analytic diffeomorphism D:Ω→B(0,ϵ) such that
∫{x∈U;φ(x)≤e}dx=∫{x∈B(0,ϵ);|x|2≤e-e0}{}Jac(D-1(x))dx.
By a simple calculus we show, using polar coordinates, that the integral of the r.h.s. is equal to H(e-e0)g(e-e0). On the other hand, since ∇φ(x)≠0 for x∈Σe0∖{x0}, it follows from Lemma 2.2 that
∫{x∈O∖U;φ(x)≤e}dx
is analytic near e0. This ends the proof of the lemma.

3. Lower-Bound Near Singularities due to Band Crossing

Here we are interested in the C∞ singular support (which will be denoted by sing supp). Recall that x0∉singsuppμ if and only if μ is C∞ near x0. The case of analytic singular support can be treated similarly.

In this section we study resonances near singularities of ρ(λ) generated by a band crossing. We will only consider the two-dimensional case. With similar assumptions, one can treat the case n≥2.

We assume that λj(k) is double eigenvalues λj-1(k0)<λj(k0)=e0=λj+1(k0)<λj+2(k0) and that for all k≠k0 such that λi(k)=e0, λi(k) is simple and ∇λi(k)≠0.

Since P0(k) is analytic in k, this implies that, for |k-k0|≤δ (with δ small enough), the span V(k), of the eigenvectors of P0(k) corresponding to eigenvalues in the set {e;|e-e0|≤δ}, has a basis ψj(x,k), ψj+1(x,k), which is orthonormal and real analytic in k. The restriction of P0(k) to V(k) has the matrix(α(k)b(k)¯b(k)β(k)),
which can be written(a(k)+c(k)b1(k)-ib2(k)b1(k)+ib2(k)a(k)-c(k)),
where a(k)=α(k)+β(k)/2, c(k)=α(k)-β(k)/2, b1(k) and b2(k) are real valued. Next the periodic potential is assumed to have the symmetry V(x)=V(-x). This symmetry is typical of metals. This symmetry forces b(k) to be real valued (i.e., b2(k)=0), (see [19]). Consequently, near k0 we haveλj(k)=a(k)-c2(k)+b12(k),λj+1(k)=a(k)+c2(k)+b12(k).
We assume that ∇b1(k0), ∇c(k0) are independent. Since n=2, (∇b1(k0),∇c(k0)) is a basis in ℝ2. Set ∇a(k0)=α1∇b1(k0)+α2∇c(k0).

Lemma 3.1.

Let ∇a(k0)=α1∇b1(k0)+α2∇c(k0) be as above. One assumes that
α12+α22<1.
Then there exist an open connected neighborhood J of e0 and analytic functions f and g such that
ρ(e)=f(e)+(H(e-e0)-H(e0-e))g(e),
with
g′′(e0)≠0,∀e∈J.

Proof.

To simplify the notation we assume that k0=0 and e0=0.

Let Ω be a neighborhood of k0=0. We introduce(2π)nρ1(e)=∫{k∈Ω;λn(k)≤e}dk+∫{k∈Ω;λn+1(k)≤e}dk,
so that
(2π)n(ρ(e)-ρ1(e))=∑j∉{n,n+1}∫{k∈E*;λj(k)≤e}dk+∫{k∈E*∖Ω;λn(k)≤e}dk+∫{k∈E*∖Ω;λn+1(k)≤e}dk.
Due to Lemma 2.2, the right-hand side of the above equalities is analytic near 0.

Since ∇b1(k0), ∇c(k0) are independent, there exist a neighborhood Ω of k0=0, ϵ>0 and a local analytic diffeomorphism κ:Ω→B(0,ϵ) such that, with the change of variable k→κ(k), we obtain(2π)nρ1(e)=∫{|k|≤ϵ;G(k)+|k|≤e}F(k)dk+∫{|k|≤ϵ;G(k)-|k|≤e}F(k)dk,
where G(k)=a(κ-1(k)) and F(k)=Jac(κ(k)) are analytic near k=0 and ∇G(0)=(α1,α2).

Using polar coordinates and making the change r→-r, ω→-ω in the second integral, we get(2π)nρ1(e)=∫S1∫{0≤r≤δ;G(rω)+r≤e}F(rω)rdrdω-∫S1∫{-δ≤r≤0;G(rω)+r≤e}F(rω)rdrdω,
which can be written
(2π)nρ1(e)=∫S1∫{0≤r≤δ;G(rω)+r≤e}F(rω)rdrdω+∫S1∫{-δ≤r≤0;G(rω)+r≥e}F(rω)rdrdω-c0,
where c0=∫S1∫{-δ≤r≤0}F(rω)rdrdω. Since
∂r(G(rω)+r)|r=0=〈∇G(0),ω〉+1≥η>0,
uniformly on ω∈S1, there exist δ1,δ2>0 (independent on ω∈S1) such that Y:r→Y(r)=G(rω)+r from ]-δ1,δ1[ into ]-δ2,δ2[ is an analytic diffeomorphism. Hence, for |e| small enough
(2π)nρ1(e)+c0=∫S1∫{t≥0;t≤e}F(Y-1(t)ω)Y-1(t)Y′(t)dtdω+∫S1∫{t≤0;t≥e}F(Y-1(t)ω)Y-1(t)Y′(t)dtdω=(H(e)-H(-e))g(e),
where
g(e)=∫0e∫S1F(Y-1(t)ω)Y′(t)Y-1(t)dtdω.
Using that
Y-1(0)=0
we deduce g′′(0)=F(0)∫S1(〈∇G(0),ω〉+1)-2dω≠0.

We denote by #A the number of elements of A, counted with their multiplicity. The main result of this section is the following.

Theorem 3.2.

Let λ,e0∈σ(P0) with λ∈(e0+singsupp(μ)). One assumes the following.

The periodic potential V satisfies V(x)=V(-x).

There exists k0∈ℝn/Γ* such that λj-1(k0)<λj(k0)=e0=λj+1(k0)<λj+2(k0).

For all k∉k0+Γ* such that λi(k)=e0, the eigenvalue λi(k) is simple and ∇λi(k)≠0.

The numbers (α1,α2) satisfy (3.4), and (λ-supp(μ))⊂J. Here J is the interval given by Lemma 3.1.

λ satisfies (H1).

Then for all h-independent complex neighborhoods Ω of λ, there exist h0=h(Ω)>0 sufficiently small and C=C(Ω)>0 such that, for h∈]0,h0[,#{z∈Ω;z∈Res(P(h))}≥CΩh-n.

Proof.

Without any loss of generality we may assume that e0=0. Set
K(⋅):=(H(⋅)-H(-⋅))g(⋅),
where g(·) is the function given in Lemma 3.1.

The assumption that (λ-supp(μ))⊂J ensures that, in the study of dρ*μ near λ, one only needs the value of ρ in J given by (3.4). More precisely, it implies that
ω(t)=dρ*μ(t)=ρ*dμ(t)=f*dμ+K(⋅)*dμ=(1)+(2),
for t near λ.

Since f is smooth, the first term of the right-hand side of the above equation is also smooth.

Clearly, it follows from assumption (2.6) and Lemma 2.2 that the singsupp(μ) is a discrete set. Thus, the point λ is isolated in singsupp(μ). We recall that we have assumed that e0=0.

Let χ∈C0∞(B(0,1)) (resp., θ∈C0∞(B(λ,1))) be equal to one near zero (resp., λ). Here B(y,r) is the disc of center y and radius r. Set χϵ=χ(·/ϵ) and θϵ=θ(·/ϵ). We choose ϵ>0 small enough such that
singsupp(μ)∩suppθϵ={λ}.

To study the second term of the right-hand side of (3.18), we write it in the form
(2)=K(⋅)(1-χϵ)*dμ+K(⋅)χϵ*θϵdμ+K(⋅)χϵ*(1-θϵ)dμ=(3)+(4)+(5).
Since K(·)(1-χϵ) is smooth the term (3) is also smooth. Using (3.19) and the fact that the support of K(·)χϵ is small for ϵ≪1, we see that the term (5) is C∞ near λ.

Now, we claim thatsingsupp(4)={λ}.
First, from a standard result on the singular support, we have
singsupp(4)⊂singsupp(K(⋅)χϵ)+singsupp(θϵdμ)={λ}.
Consequently, to prove the claim it suffices to show that (4)∉C0∞(R). We recall that (4) has a compact support.

A simple calculus and Lemma 3.1 yieldc(1+|ξ|2)-1≤|K(⋅)χϵ̂(ξ)|≤C.
Here f̂(ξ) is the Fourier transform of f. Consequently, θϵdμ̂∈S(ℝ) if and only if (4)̂∈S(ℝ), where S(ℝ) is the Schwartz space of C∞ function of rapid decrease.

On the other hand, (3.19) implies that θϵμ̂∉S(ℝ). Combining this with the above remarks we get the claim.

Summing up, we have proved that λ∈singsupp(ω=dρ*μ).

Now, applying the following result of [14] we obtain Theorem 3.2.Theorem 3.3 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

Let λ∈sing suppa(ω). Assume that λ satisfies (H1). Then for every h-independent complex neighborhood Ω̃ of λ, there exists h0=h(Ω̃) sufficiently small and C=C(Ω̃) large enough such that, for h∈]0,h0[,#{z∈Ω̃;z∈Res(P(h))}≥C(Ω̃)h-n.

Remark 3.4.

Let e0 be a singularity of the integrated density of states, generated by a band crossing. Theorem 3.2 shows that there is at least ~h-n resonances near e0+t, where t is in the singular support of the distribution μ defined byμ(t)=∫{x∈Rn;W(x)>t}dx.

4. Lower Bound of the Counting Function near the Edges of Bands

In this section we study resonances generated by analytic singularities of ρ near the edge of bands. The following result is a consequence of Lemma 2.3.

Lemma 4.1.

Let e0∈σ(P0). One assumes the following.

If λj(k)=e0, then λj(k) is a simple eigenvalue of H0(k).

There exist i0 and k0 such that λi0(k0)=e0, ∇λi0(k0)=0, ±∂2λi0(k0)>0 and ∇λi0(k)≠0,for allk∈E*,k≠k0.

For all k∈λi-1{e0} with i≠i0, ∇λi(k)≠0.

Then there exists an open connected neighborhood J of e0 such that
ρ(e)=f(e-e0)+H(±(e-e0))g(±(e-e0)),∀e∈J,
where f and g are analytic near zero and g(0)=0,…,g(n-1)(0)=0,g(n)(0)≠0. Here, +(-) corresponds to a local minimum (maximum, resp.).

Now, repeating the arguments in the proof of Theorem 3.2 and using Lemma 4.1, we obtain the following.

Theorem 4.2.

Let e0,λ∈σ(P0) with λ∈(e0+sing suppa(μ)). One assumes the following.

λ satisfies (H1),

e0 satisfies the assumptions of Lemma 4.1,

(λ-supp(μ))⊂J. Here J is the interval given by Lemma 4.1.

Then for all h-independent complex neighborhoods Ω of λ, there exist h0=h(Ω)>0 sufficiently small and C=C(Ω)>0 such that, for h∈]0,h0[,
#{z∈Ω;z∈Res(P(h))}≥CΩh-n.Remark 4.3.

Notice that the assumptions (iv) in Theorem 3.2 and (iii) in Theorem 4.2 are satisfied if ∥W∥∞ is small.

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