# Resonances for Perturbed Periodic Schrödinger Operator

Authors: Mouez Dimassi
Journal: Advances in Mathematical Physics
 (2012)

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## Abstract

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


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## Body

## 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 equation(1.1)P(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
(1.2)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 Hamiltonian(1.3)P0=-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 whereV=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 (1.4)μ(t)=∫{x∈Rn;W(x)>t}dx,ρ0(t)=(2π)-nvol(B(0,1))(max⁡(t,0))n/2.Using the explicit formula ofρ0 we see that the analytic singular support of the distributions μ and dρ0*μ coincide.In the case whereV≠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
(1.5)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).Ifλj(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 Section3 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

LetΓ=⊕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*).LetV be a real valued potential, C∞ and Γ-periodic. For k in ℝn, we define(2.1)P0(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:(2.2)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 thatλn(k) is a continuous function of k and is real analytic in a neighborhood of any k such that(2.3)λn-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. (H1)
For allk0 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 withP0. The density of states measure ρ is defined as follows:(2.4)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:(2.5)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(2.6)|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 ofP(h) by the spectral deformation method (see [17]). We follow essentially the presentation of [13].Letv∈C∞(ℝn;ℝn) be Γ*-periodic. For t∈ℝ, we introduce the spectral deformation family 𝒰t defined by for all u∈𝒮:(2.7)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:(2.8)[Fhu](ξ):=∫Rne-(i/h)xξu(x)dx,∀u∈S(Rn).
Consider, for t∈ℝ, the family of unitarily equivalent operators(2.9)P1(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)).Forf∈C0∞(ℝ), we set
(2.10)〈μ,f〉=∫[f(W(x))-f(0)]dx,(2.11)〈ω,f〉=1(2π)n∑j≥1∫E*∫Rxn[f(W(x)+λj(k))-f(λj(k))]dkdx,
For E>0, let(2.12)ν+(E):=∫{x∈Rn;W(x)≥E}dx.
Similarly, for E<0, we set(2.13)ν-(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) and(2.14)μ|R±=-ddEν±(E).Lemma 2.1.
The distributionsω and μ are real valued of order ≤1. Moreover, in𝒟′(ℝ), one has(2.15)ω=dρ*μ.Proof.
Applying Taylor’s formula to the right-hand side of (2.10), we obtain
(2.16)|〈μ,f〉|≤sup⁡|f′|∫|W(x)|dx,
which together with (2.6) imply that μ is a distribution of order ≤1, with
(2.17)supp⁡μ⊂[inf⁡W(x),sup⁡W(x)].
Consequently, dρ*μ is well defined in 𝒟′(ℝ) and for all f∈C0∞(R), we have
(2.18)〈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.LetΩ 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 function(2.19)I(e):=∫{x∈Ω;φ(x)≤e}dx.
For e∈φ(Ω), we set(2.20)Σ(e):={x∈Ω;φ(x)=e}.Lemma 2.2.
Lete0∈φ(Ω), and let  Σ(e),  I(e) be as above. One assumes that (i)
∇φ(x)≠0 for all x∈Σ(e0),(ii)
∂Ω∩Σ(e0)=∅.
Then the function(2.21)I(e):=∫{x∈Ω;φ(x)≤e}dx
is analytic near e0.Proof.
Letϵ 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
(2.22)∫{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.
Ifφ has a nondegenerate extremum at  x0 with  φ(x0)=e0 and if  ∇φ(x)≠0 for all  x∈Σe0∖{x0}, then
(2.23)I(e)=f(e-e0)+H(±(e-e0))g(±(e-e0)),
where f and g are analytic near zero and
(2.24)g(t)~t→0vol(Sn-1)ndet⁡φ′′(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
(2.25)∫{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
(2.26)∫{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 theC∞ 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 ofρ(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 thatλj(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.SinceP0(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(3.1)(α(k)b(k)¯b(k)β(k)),
which can be written(3.2)(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 have(3.3)λj(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
(3.4)α12+α22<1.
Then there exist an open connected neighborhood J of e0 and analytic functions f and g such that
(3.5)ρ(e)=f(e)+(H(e-e0)-H(e0-e))g(e),
with
(3.6)g′′(e0)≠0,∀e∈J.Proof.
To simplify the notation we assume thatk0=0 and e0=0.
LetΩ be a neighborhood of k0=0. We introduce(3.7)(2π)nρ1(e)=∫{k∈Ω;λn(k)≤e}dk+∫{k∈Ω;λn+1(k)≤e}dk,
so that
(3.8)(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(3.9)(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 changer→-r, ω→-ω in the second integral, we get(3.10)(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
(3.11)(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
(3.12)∂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
(3.13)(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
(3.14)g(e)=∫0e∫S1F(Y-1(t)ω)Y′(t)Y-1(t)dtdω.
Using that
(3.15)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.
Letλ,e0∈σ(P0) with  λ∈(e0+singsupp(μ)). One assumes the following. (i)
The periodic potentialV satisfies V(x)=V(-x).(ii)
There existsk0∈ℝn/Γ* such that λj-1(k0)<λj(k0)=e0=λj+1(k0)<λj+2(k0).(iii)
For allk∉k0+Γ* such that λi(k)=e0, the eigenvalue λi(k) is simple and ∇λi(k)≠0.(iv)
The numbers(α1,α2) satisfy (3.4), and (λ-supp(μ))⊂J. Here J is the interval given by Lemma 3.1.(v)
λ satisfies (H1).
Then for allh-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[,(3.16)#{z∈Ω;z∈Res(P(h))}≥CΩh-n.Proof.
Without any loss of generality we may assume thate0=0. Set
(3.17)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
(3.18)ω(t)=dρ*μ(t)=ρ*dμ(t)=f*dμ+K(⋅)*dμ=(1)+(2),
for t near λ.
Sincef 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.
Letχ∈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
(3.19)singsupp⁡⁡(μ)∩supp⁡θϵ={λ}.
To study the second term of the right-hand side of (3.18), we write it in the form
(3.20)(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 that(3.21)singsupp⁡⁡(4)={λ}.
First, from a standard result on the singular support, we have
(3.22)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 Lemma3.1 yield(3.23)c(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 thatλ∈singsupp(ω=dρ*μ).
Now, applying the following result of [14] we obtain Theorem 3.2.Theorem 3.3 (see [14]).
Letλ∈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[,(3.24)#{z∈Ω̃;z∈Res(P(h))}≥C(Ω̃)h-n.Remark 3.4.
Lete0 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 by(3.25)μ(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 ofρ near the edge of bands. The following result is a consequence of Lemma 2.3.Lemma 4.1.
Lete0∈σ(P0). One assumes the following. (i)
Ifλj(k)=e0, then λj(k) is a simple eigenvalue of H0(k).(ii)
There existi0 and k0 such that λi0(k0)=e0,  ∇λi0(k0)=0, ±∂2λi0(k0)>0 and ∇λi0(k)≠0,for allk∈E*,k≠k0.(iii)
For allk∈λi-1{e0} with i≠i0, ∇λi(k)≠0.Then there exists an open connected neighborhood J of e0 such that
(4.1)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 Theorem3.2 and using Lemma 4.1, we obtain the following.Theorem 4.2.
Lete0,λ∈σ(P0) with  λ∈(e0+sing suppa(μ)). One assumes the following. (i)
λ satisfies (H1),(ii)
e0 satisfies the assumptions of Lemma 4.1,(iii)
(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[,
(4.2)#{z∈Ω;z∈Res(P(h))}≥CΩh-n.Remark 4.3.
Notice that the assumptions (iv) in Theorem3.2 and (iii) in Theorem 4.2 are satisfied if ∥W∥∞ is small.



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