Periodic Jacobi operator with finitely supported perturbation on the half-lattice

Periodic Jacobi operator with finitely supported

perturbation on the half-lattice

Alexei Iantchenko

WIMCS and Aberystwyth University, Wales, UK, on leave from Malm¨o, Sweden

6 april 2010

History

A lot of papers is devoted to the resonances for the Schr¨odinger operator −_dxd22+ q(x ) on the line R and half-line with compactly supported perturbation: R.Froese, B.Simon, M.Zworski, E.Korotyaev,...

Inverse resonance problem was solved.

The problem of resonances for the Schr¨odinger with periodic plus compactly supported potential −_dxd22 + p(x ) + q(x ) is much less studied: N.Firsova, F.Klopp and M.Marx, E.Korotyaev. (Spectral properties of the perturbed Hill operators are much better studied)

Inverse resonance problem is not yet solved. Finite-difference Schr¨odinger and generalizations express many similar features and are simpler, in the sense that the spectrum is finite and we work with polynomials.

Spectral properties of infinite Jacobi matrices are much studied: M. Toda, J.S.Geronimo and W.Van Asshe and Case, D.Damanik and B.Simon,... Inverse problem is solved for periodic Jacobi: L.Percolab (1984), ?

Scattering problem for asymptotically periodic Jacobi matrices was extensively studied by J.Geronimo, K.Case, M.Kac (1973,1980), G.Guseinov (1976), D.Damanik and B.Simon, N.Firsova, M.Toda, V.A.Marchenko, ...

Inverse scattering problem was solved by Ag.Kh. Khanmamedov (6 2005 line, 2006 half-line) and I.Egorova, J.Michor, G.Teschl (2006 line, quasi-periodic background). Resonance problems are less studied: M.Marletta and R.Weikard (2007), E.Korotyaev (arxiv, constant background, line).

The present work is the part of the joint project with E. Korotyaev. Applications: resonances for nanotubes and inverse resonance scattering for Schr¨odinger.

Half-infinite Jacobi matrix

J =                    ˜ b1 ˜a1 0 0 ... ˜ a1 ˜b2 ˜a2 0 ... 0 ˜a2 ˜b3 ˜a3 ... 0 0 ˜a3 ˜b4 ... 0 0 0 ˜a4 ... ... ... ... ... ...                   

in `2(N) and associated with Jacobi operator

(Jy )n= ˜an−1yn−1+ ˜anyn+1+ ˜bnyn, Dirichlet b.c. y0= 0, (1)

˜

an= an+ ˘an, ˜bn= bn+ ˘bn∈ R,

an= an+q> 0, bn= bn+q∈ R, n, q ∈ N, periodic with period q > 2, (2)

˘

Operator J is a finitely supported perturbation of the operator J0_{defined by (1) with}

coefficients ˜aj= aj, ˜bj= bj, j ∈ N, verifying (4), i.e. periodic with period q on N. We

will use the standard notation {cn}n∈N≡ c.

Let ϕ = (ϕn(λ))n∈Zand ϑ = (ϑn(λ))n∈Zbe fundamental solutions for equation

an−1yn−1+ anyn+1+ bnyn= λyn, λ ∈ C, (4)

under the conditions ϑ0= ϕ1= 1 and ϑ1= ϕ0= 0. Let ψ±= ϑ + m±ϕ ∈ `2(Z±) be

the Floquet-Bloch functions. Here m±=

1

2(ϕq+1−ϑq)±

√

∆2−1

ϕq are the

Titchmarch-Weyl functions, ∆(λ) = 2−1(ϕq+1+ ϑq) the Lyapunov function.

The perturbation (˘a, ˘b) satisfying (3) does not change the absolutely continuous spectrum: σac(J) = σac(J0) =Sqn=1[λ

+ n−1, λ

−

We fix the branch of the functionp∆2_{(λ) − 1 on C by demandingp∆}2_{(λ) − 1 < 0}

for λ < λ+₀. Now we introduce the two-sheeted Riemann surface Λ ofp∆2_{(λ) − 1}

obtained by joining the upper and lower rims of two copies of the cut plane

Γ = C \ σac(J0) in the usual (crosswise) way. The k−th gap on the first physical sheet

Λ+ we will denote by γ_k+= (λ−_k, λ+_k), k = 1, . . . , q − 1,

γ₀+= (−∞, λ+₀), γ+ q = (λ

−

q, ∞) infinite gaps,

and the same gap but on the second nonphysical sheet Λ−we will denote by γ_k−and

let γc k= γ + k ∪ γ − k.

Let R(λ) = (J − λ)−1_{denote the resolvent and h·, ·i the scalar product in `}2_(N).

Definition. 1) A number λ0∈ Λ−is a resonance, if the function hRf , g i has a pole at

λ0for some f , g ∈ `2comp(N). The multiplicity of the resonance is the multiplicity of

the pole. If Re λ0= 0, we call λ0antibound state.

2) A real number λ0such that (∆(λ0))2− 1 = 0 is a virtual state if hRf , g i has a

singularity at λ0for some f , g ∈ `2comp(N).

3) The state λ ∈ Λ is a bound state or a resonance or a virtual state of J.

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−3 −2 −1 0 1 2 3 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

Let fn±be the Jost solutions, fn±= ˜ϑn+ m±ϕ˜n, where ˜ϑn, ˜ϕndenote the solutions to

˜an−1yn−1+ ˜anyn+1+ ˜bnyn= λyn, satisfying ˜ϑn= ϑn, ˜ϕn= ϕn, for n > p. The

functions ˜ϕ, ˜ϑ are polynomials, the Jost solutions f±and Titchmarch-Weyl functions m±are meromorphic functions on Λ. The projection of all singularities of m±to the

complex plane coincides with the set of zeros {µk}q−1k=1 of polynomial ϕq.

If an= 1, bn= 0 then J0is discreet Laplacian and all states of J (eigenvalues,

resonances) are poles of the scattering matrix S(λ) =f − 0 (λ)

f₀+(λ). If J0is periodic on N

then there are real states (bound, antibound or virtual) which are not zeros of the Jost function, namely µ+

n or µ −

n projected on the Dirichlet eigenvalue µn. Then

Direct problem

Theorem 1. Let µn∈ γndenote the Dirichlet eigenvalue: ϕq(µn) = 0.

i) The point λ ∈ ∪k=0,...,qγk+∪ Λ−is a state of J iff one of the following two

conditions is satisfied:

1) πCλ06∈ {µn}q_k=1and f₀+(λ0) = 0;

2) π_Cλ0= µnfor some n = 1, . . . , q − 1 and ˜ϕ0(πCλ0) = 0.

If λ0, with πCλ0= µnis a bound or an antibound state then it is necessarily simple.

ii) If λ1∈ Λ+is a bound state then λ26∈ σst(J), where λ2∈ Λ−is the same number

Theorem 2.

Suppose ˘bp6= 0. Then Jacobi operator J satisfying (1–3) has

either 2p + q − 1 states (if ˘ap6= 0) or 2p + q − 2 states (if ˘ap= 0)

counted with multiplicities. Moreover the following facts hold true. 1) The total number of bound and virtual states is > 2.

2) In the closure of the gap γ_kc= γ_k+∪ γ−_k, k = 1, . . . , q − 1, there is always an odd number of states with at least one bound or virtual state.

3) For any k = 0, . . . , q, let λ1< λ2be any two bound states of J, λ1,2∈ γ_k+, such

that there are no other eigenvalues on the interval Ω+_{= (λ}

1, λ2) ⊂ γ+k. Then there

exist an odd number > 1 of antibound states on Ω−_{, where Ω}−_{⊂ γ}−

k ⊂ Λ−is the

same interval but on the second sheet, each antibound state being counted according to its multiplicity.

Inverse problem

We suppose that all gaps are open: λ−_k < λ+_k, k = 1, . . . , q − 1, and define the class of finite support perturbations: let p ∈ N, κ ∈ {2p + q − 2, 2p + q − 1} and

X2p+q−1= n (˘a, ˘b) = (˘an, ˘bn)∞n=0∈ (`2(N))2, an+ ˘an> 0, ˘bn∈ R : ˘ aj= ˘bj= 0, for j > p, ˘bp6= 0, ˘ap6= 0 o , (5) X2p+q−2= n (˘a, ˘b) = (˘an, ˘bn)∞n=0∈ (`2(N))2, an+ ˘an> 0, ˘bn∈ R : ˘ aj= ˘bj= 0, for j > p, ˘bp6= 0, ˘ap= 0 o . (6)

Quasi-momentum map

We consider the quasi-momentum map which is bijection

ω(λ) = ei κ(λ)_{: C}±_{7→ W}1_{(±) = {ω ∈ C : |ω| < 1, ∓ Im ω > 0} \}Sq−1 k=1β1k(∓), where β1 k(±) = h ω(αk), ω(λ−k+1) i = {ω ∈ C : arg ω = ±kπ_q ; e−hk_{6 |ω| 6 1}.} Denote W1= W1(−) ∪ W1(+) ∪ (−1, 1). Here is example for q = 3.

Quasi-momentum map

We consider the quasi-momentum map which is bijection

ω(λ) = ei κ(λ)_{: C}±_{7→ W}1_{(±) = {ω ∈ C : |ω| < 1, ∓ Im ω > 0} \}Sq−1 k=1β 1 k(∓), where β1 k(±) = h ω(αk), ω(λ−k+1) i = {ω ∈ C : arg ω = ±kπ_q ; e−hk_{6 |ω| 6 1}.} Denote W1_{= W}1_{(−) ∪ W}1_{(+) ∪ (−1, 1).} Put W2(±) = {ω ∈ C : |ω| > 1, ∓ Im ω > 0} \Sq−1 k=1β 2 k(∓), where β2 k(±) = h (ω(αk))−1, ω(λ−k+1) i =nω : arg ω = ±kπ_q ; 1 6 |ω| 6 ehk o . Let W2_{= W}2_{(−) ∪ W}2_{(+) ∪ (−∞, −1) ∪ (1, ∞).}

By identifying the symmetric points of the corresponding sides of the cuts β_kj(−) and βj_k(+), j = 1, 2, we obtain Riemann surfaces W1,2which are isomorphic to Λ±.

The Jost solutions ψ±, f±are meromorphic in W1,2and continuous up to the boundary with the only possible singularities at ω(µn) and at 0.

Let Rκdenote a set of vectors {ω(λj)}κ1 such that λj∈ σbs(J), j = 1, . . . , N,

λ1< λ2< . . . < λN, are all bound states of J. We show that Rκis the set of all zeros

of a real polynomial p(ω) of order κ and describe the characteristic properties. We introduce a class of vectors Zκsuch that for any (˘a, ˘b) ∈ Xκwe have inclusion

Rκ⊂ Zκ.

For κ ∈ {2p + q − 2, 2p + q − 1} we construct the mapping:

F: Xκ→ Zκ, (˘a, ˘b) → (ωj(˘a, ˘b))κj =1∈ Zκ, (7)

i.e., to each (˘a, ˘b) ∈ Xκwe associate a vector ω = (ωn)κ1 ∈ Zκ.

We prove the following results:

i) The mapping F : Xκ→ Zκis one-to-one and onto. In particular, a pair of

coefficients in Xκis uniquely determined by its states.

The inverse problem for mapping F can be divided into three parts:

1. Uniqueness. Do the states determine uniquely (˘a, ˘b) ∈ Xκ?

2. Reconstruction. Give an algorithm for recovering (˘a, ˘b) from {ωj}κj =1∈ Zκonly.

3. Characterization. Give necessary and sufficient conditions for {ωj}κj =1to be the

Zigzag nanotube in magnetic field

We consider the Schr¨odinger operator Hb _{on the zigzag nanotube Γ ⊂ R}3_(1D

tight-binding model of zigzag single-wall nanotubes) in a uniform magnetic field B= |B|e0, e0= (0, 0, 1) ∈ R3and in an external electric potential.

Our carbon model nanotube, simply, is the honeycomb lattice of a graphene sheet rolled into a cylinder.

Tight-binding approximation

Singlewall nanotubes, one atomic layer in thickness in the radial direction, are the most likely candidate for miniaturizing electronics beyond the micro electromechanical scale that is currently the basis of modern electronics.

In the tight binding model for a solid-state lattice of atoms, it is assumed that the full Hamiltonian H of the system may be approximated by the Hamiltonian of an isolated atom centered at each lattice point.

Hamiltonian

Let Z = Z × {0, 1} × ZN, ZN= Z/(NZ). Introduce the Hilbert space `2(Γ) of

functions f = (fω)ω∈Z on Γ equipped with the norm kf k2_`2_(Γ)= P

ω∈Z|fω|2.

The tight-binding Hamiltonian Hb _{on the nanotube Γ has the form H}b_{= H}b 0+ V on

`2_{(Γ), where H}b

0 is the Hamiltonian of the nanotube in the magnetic field:

(H0bf )n,0,k= eib1fn−1,1,k+ eib2fn−1,1,k+1+ eib3fn,1,k, (H0bf )n,1,k = eib1fn+1,0,k−1+ eib2fn+1,0,k+ e−ib3fn,0,k, f = (fω)ω∈Z, ω = (n, j , k) ∈ Z, b3= 0, b1= −b2= b = 3|B| 16 cot π 2N, and the operator V corresponding to the external electric potential is given by (Vf )ω= Vωfω, where Vn−1,1,k= v2n, Vn,0,k= v2n+1, v = (vn)n∈Z∈ `∞.

Direct sum of Jacobi operators

Each operator Hb_{, b ∈ R is unitarily equivalent to the operator ⊕}N

1Jkb, where Jkb is a

Jacobi operator, acting on `2(Z) and given by

(Jkby )n= an−1yn−1+ anyn+1+ vnyn, y = (yn)n∈Z∈ `2,

a2n= 2|ck|, a2n+1= 1, ck= cos(b +

πk

Jacobi on half-line: Z

+

We consider Jacobi operator on Z+. Let v ∈ R, p ∈ Z+, and consider J ≡ Jkb:

(J_kby )n= an−1yn−1+ anyn+1+ ˜vnyn, y = (yn)n∈Z+∈ `

2_,

where a2n= a ≡ 2|ck|, a2n+1= 1, ck= cos(b +πkN), n ∈ Z and

˜

vj= vj+ qj, v2n+1= v , v2n= −v , qj= 0, for j > p.

Perturbation of 2-periodic Jacobi: J = J0+ q,

n = 0, 1, 2, . . . , (J0y )n= an−1yn−1+ anyn+1+ vnyn, a2n+1= 1, a2n= a ∈ [0, 2],

v2n+1= v , v2n= −v .

Meaning of perturbation potential

I Effective potential arising from from the boundary

Our results

We classify all states of J (virtual, bound, anti-bound, complex resonances) and describe how the spectrum of J and resonances vary when the magnetic field b, electric field v and perturbation qkchange.

Generalized Jost solutions

The generalized Jost solutions are the fundamental solutions fn±to the equation

Jyn= an−1yn−1+ anyn+1+ ˜vnyn= λyn, (λ, n) ∈ C × Z+, ˜vn= vn, for n > p, (8) satisfying fn+= ψn+, f − n = ψ − n, for n > p.

Here ψ±n are Jost solutions - the Floquet solutions

ψ₀±= 1, ψ₁±= m±, ψ₂±= e±2i κ= ξ±2, ψ±2n= ξ 2n

±, ψ±2n+1= ξ 2n ±m±,

where ξ±2 = e±2i κare Floquet multipliers.

The Weyl-Titchmarch function m±=∆ + a ±

p∆2_{(λ) − 1}

Generalized Jost solutions

The functions fn±have the form

fn±= ˜ϑn+ m±ϕ˜n, m±=

∆ + a ±p∆2_{(λ) − 1}

λ − v , (9)

where ˜ϑ(·), ˜ϕ(·) are polynomials, satisfying

˜

ϑn(λ) = ϑn(λ), ˜ϕn(λ) = ϕn(λ) for n > p.

The functions fn±are meromorphic on the Riemann surface Λ ' Λ+∪ Λ−,

Λ±= C \ σac(J0). Each sheet is a copy of the complex plane slit along σac. The first

sheet Λ+_{is glued to the second sheet Λ}−_{by identifying “crosswise” the slits in}

σac(J0). We denote the copies of γ0,1,2 on Λ+ (respectively Λ−), by γ+0,1,2

(respectively γ_0,1,2− ).

Resolvent

Let {φn, ψn} = an(φnψn+1− φn+1ψn} denote the Wronskian. The kernel of the

resolvent of J is

R(n, m) = hen, (J − λ)−1emi = −

Φnfm+

{Φ, f+_}, n < m,

where en= (δn,j)j ∈Z, JΦn= λΦn, Φ0= 0, Φ1= 1, andΦ, f+ = −a0f0+.

Each function Φn(λ), n ∈ N, is polynomial in λ.

The function R(n, m) is meromorphic on Λ for each n, m ∈ Z. The singularities of R(n, m) are given by the singularities of

Rn(λ) = f+ n(λ) f₀+(λ) = ˜ ϑn(λ) + m+(λ) ˜ϕn(λ) f₀+(λ) .

Classification of states

The state λ ∈ Λ is a bound state, resonance or virtual state.

1) A value λ0∈ γ_k+, k = 0, 1, 2, is a bound state, if the function Rn(λ) has a pole at

λ0for almost all n ∈ Z+.

2) A value λ0∈ Λ−, is a resonance, if the function Rn(λ) has a pole at λ0for almost

all n ∈ Z+. If Re λ0= 0, we call λ0antibound state.

3) A value λ0= λ±0 or λ0= λ±1 is a virtual state if (Rn(λ))2or Rn(λ) has a pole at

λ0for almost all n ∈ Z+.

Function F

The function F (λ) = ϕ2f0+f −

0 is polynomial of degree 6 2p and we have

the set of states of J is the set of zeros of F .

F = ϕ2ϑ˜20+ 2φ ˜ϑ0ϕ˜0− ϑ3ϕ˜20= (λ − v ) ˜ϑ20+

1 a(λ

2_{− v}2_{+ a}2_{− 1) ˜ϑ}

0ϕ˜0+ (λ + v ) ˜ϕ20. (10)

If λ1∈ γk+, k = 0, 1, 2, is bound state and λ16= v then (−1)kF0(λ1) < 0.

Hence all bound states 6= v of J are simple and between any two bound states λ1,3∈ γk(not separated by a band of the absolute continuous spectrum) there is at

least one antibound state λ2and (−1)kF0(λ2) > 0.

Moreover, for any λ ∈ σ(J0), different from any of the endpoints λ±_0,1, F (λ) is strictly positive above the right band of σac(J0) and F (λ) is strictly negative below the left

For p = 7, v = [1; 1; 1; 1.2; 0.2; 0.2; 0.2] v = −0.5 and a = 2|cos(b + πk/N)|, where k = 1, N = 1, and b varies from b = 0.8 to b = −3, with step −0.2 (sequence of 20 pictures, covering the whole period of a = a(b)),

we draw the zeros of the function F (λ) (upper picture) and profile of function y = F (λ) for λ ∈ R. Green circles on the upper picture shows that the encircled zero of F is the zero of the Jost function f₀+on Λ+, i.e. the eigenvalue.

Recall that the total number of zeros of F is 2p which is 14 in this case. In the first picture we plot the values of a.

0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Values of a for p=7; [v 1;...;vp]=[1;1;1;1.2;0.2;0.2;0.2]; v=−0.5

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2 3 4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −4 −3 −2 −1 0 1 2 3 4 −2 −1 0 1 2

Special cases and limit a → 0

The absolute continuous spectrum σac(J) =

σac(J0) = [− q v2_{+ (a + 1)}2_{, −} q v2_{+ (a − 1)}2_{] ∪ [} q v2_{+ (a − 1)}2_, q v2_{+ (a + 1)}2_] = [−pv2_{+ 1 −√} a v2_{+ 1}+ O(a 2_{), −}p v2_{+ 1 +√} a v2_{+ 1}+ O(a 2_)]∪ ∪ [pv2_{+ 1 −√} a v2_{+ 1}+ O(a 2_),p v2_{+ 1 +√} a v2_{+ 1}+ O(a 2_)] −→a→0 {− p v2_{+ 1} ∪ {}p v2_{+ 1}.}

If p = 1, there are two bound states

λ1,2= −[(v − ˜v1)2+ a2] ± q (a2_{− (˜v}2 1− v2))2+ 4(˜v1− v )2 2(v − ˜v1) =−(v − ˜v1) ±p(v + ˜v1) 2_{+ 4} 2 + O(a 2_{) = ±}q_{[v + (q} 1/2)]2+ 1 + q1/2 + O(a2), where ˜v1= v + q1.

Special cases and limit a → 0

σac(J0) = [− p v2_{+ 1 −√} a v2_{+ 1}+ O(a 2_{), −}p v2_{+ 1 +√} a v2_{+ 1}+ O(a 2_)]∪ ∪ [pv2_{+ 1 −√} a v2_{+ 1}+ O(a 2_),p v2_{+ 1 +√} a v2_{+ 1}+ O(a 2_)] −→a→0 {− p v2_{+ 1} ∪ {}p_v2_{+ 1}.}

If p = 2 there are two bound states

λ1,2= ˜ v1+ ˜v2±p(˜v1− ˜v2)2+ 4 2 +O(a 2_{) = ±} s  v +q1− q2 2 2 + 1+q1+ q2 2 +O(a 2_), ˜

v1= v + q1, ˜v2= −v + q;and two resonances λ3,4=

± s  ˜v1− v 2 2 −v˜1+ v ˜ v2+ v +˜v1+ v 2 +O(a 2_{) = ±} s q₁ 2 2 −2v + q1 q2 +2v + q1 2 +O(a 2_), complex if q1= 0 or if either 0 < q2< 8v +4q1 q₁2 or 8v +4q1 q2₁ < q2< 0.

Special cases and limit a → 0

The last pair are two complex resonances: Im λ3,46= 0 if q1= 0 or if either

0 < q2< 8v + 4q1 q2 1 or 8v + 4q1 q2 1 < q2< 0. (11)

This is also true for any p even. For p odd there are no complex resonances in the limit a → 0.