# Threshold Effects for the Generalized Friedrichs Model with the Perturbation of Rank One

Authors: Saidakhmat Lakaev; Arsmah Ibrahim; Shaxzod Kurbanov
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A familyHμ(p), μ>0, p∈𝕋2 of the Friedrichs models with the perturbation of rank
one, associated to a system of two particles, moving on the two-dimensional lattice ℤ2 is considered.
The existence or absence of the unique eigenvalue of the operator Hμ(p) lying below threshold
depending on the values of μ>0 and p∈Uδ(0)⊂𝕋2 is proved. The analyticity of corresponding
eigenfunction is shown.


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## Body

## 1. Introduction

In celebrated work [1] of Simon and Klaus it has been considered a family of the Schrödinger operators H=-Δ+μV and, a situation where as μ tends to μ0 some eigenvalue ei(μ) tends to 0, that is, as μ↓μ0 an eigenvalue is absorbed into continuous spectrum, and conversely, for any μ:μ>μ0 continuous spectrum gives birth to a new eigenvalue. This phenomenon in [1] is called coupling constant threshold.In [2] the Hamiltonian of a system of two identical quantum mechanical particles (bosons) moving on the d-dimensional lattice ℤd,d≥3 and interacting via zero-range repulsive pair potentials has been considered. For the associated two-particle Schrödinger operator Hμ(k),k∈𝕋d=(-π,π]d the existence of coupling constant thresholdμ0=μ0(k)>0 has been proved: the operator has none eigenvalue for any 0<μ≤μ0, but for each μ>μ0 it has a unique eigenvalue z(μ,k) above the upper threshold of the spectrum.Note that in [1] the existence of a coupling constant threshold has been assumed, at the same time in [2] the coupling constant threshold has been definitely found by the given data of the Hamiltonian.We remark that for the Hamiltonians of a system of two arbitrary particles moving onℝd or ℤd,d≥1 the coupling constant threshold vanishes, if d=1,2 and the coupling constant threshold is positive, if d≥3.Notice also that for the Hamiltonians of a system of two identical particles moving onℝd or ℤd,d=1,2 the coupling constant threshold vanishes, if particles are bosons and the coupling constant threshold is positive, if particles are fermions.In [3] for a wide class of the two-particle Schrödinger operators Hμ(k) on the d-dimensional lattice ℤd, d≥3, k being the two-particle quasimomentum, it has been proved that if the following two assumptions (i) and (ii) are satisfied, then for all k≠0, the discrete spectrum of Hμ(k) below its threshold is nonempty. The assumptions are (i) the two-particle Schrödinger operator Hμ(0), corresponding to the zero value of the quasimomentum k, has a coupling constant thresholdμ0(0)>0 and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semigroups.In [4] a family of the Friedrichs models Hμ(p),μ>0,p∈(-π,π]3 with perturbation of rank one associated to a system of two particles on the three-dimensional lattice ℤ3 has been considered. In some special case of the multiplication operator and under the assumption that the operator Hμ(0),0∈𝕋3 has a coupling constant thresholdμ0(0)>0, the existence of a unique eigenvalue, below the threshold of the spectrum of Hμ0(0)(p),p∈(-π,π]3 for all nontrivial values of p∈𝕋3, has been proved.In the present paper, a family of the Friedrichs modelsHμ(p), μ>0,p∈Uδ(0)⊂𝕋2, where Uδ(0) is a δ-neighborhood of the point p=0∈𝕋2 with perturbation of rank one associated to a system of two particles on the two-dimensional lattice ℤ2 interacting via pair local potentials, is considered and the following results have been obtained.(i)
If the parameters of the Friedrichs model satisfy some conditions (see Theorem2.3), then there exists a coupling constant thresholdμ0=μ0(p)>0: for any 0<μ≤μ0(p) the operator has none eigenvalue; at the same time for any μ>μ0(p) it has a unique eigenvalue z(μ,p), lying below its threshold of the spectrum. Moreover an explicit expression for the corresponding eigenfunction is found and its analyticity is proven.(ii)
If the parameters of the Friedrichs model do not satisfy conditions mentioned in (i), then the operator has none positivecoupling constant threshold, that is, for any μ>0 the operator Hμ(p) has a unique eigenvalue z(μ,p), lying below its threshold of the spectrum.(iii)
A criterion for being the thresholdm(p),p∈Uδ(0) of the spectrum of Hμ(p) a virtual level of the operator Hμ(p) is proven.Note that the generalized Friedrichs models appear in the problems of quantum mechanics [5], solid state physics [6], and quantum field theory [7, 8] and the existence of its eigenvalues and resonances have been studied in [9, 10].In [11] a special family of generalized Friedrichs models has been considered and the existence of eigenvalues for some values of quasimomentum p∈𝕋d of the system, lying in a neighborhood of some p0∈𝕋d, has been proved.


## 2. Notions and Assumptions: The Main Results

Letℤ be the one-dimensional hypercubic lattice and 𝕋2=(ℝ/2πℤ)2=(-π,π]2 be the two-dimensional torus, the dual group of ℤ2 (Brillion zone). Note that operations addition and multiplication by number of the elements of torus 𝕋2≡(-π,π]2⊂ℝ2 is defined as operations in ℝ2 by the module (2πℤ)2.LetL2(𝕋2) be the Hilbert space of square-integrable functions defined on the torus 𝕋2 and C1 be one-dimensional complex Hilbert space.We consider the family of generalized Friedrichs model acting inL2(𝕋2) as follows:
(2.1)Hμ(p)=H0(p)-μΦ*Φ,μ>0.
Here
(2.2)Φ:L2(T2)⟶C1,Φf=(f,φ)L2(T2),Φ*:C1⟶L2(T2),Φ*f0=φ(q)f0,
where (·,·)L2(𝕋2) is inner product in L2(𝕋2) and H0(p),p∈𝕋2 is the multiplication operator by function wp(·):=w(p,·), that is,
(2.3)(H0(p)f)(q)=wp(q)f(q),f∈L2(T2).
Note that for anyf∈L2(𝕋2) and g0∈C1 the equality
(2.4)(Φf,g0)C1=(f,Φ*g0)L2(T2),
holds.The following assumption will be needed throughout the paper.Assumption 2.1.
The following conditions are satisfied:(i)
the functionφ(·) is nontrivial real-analytic on 𝕋2;(ii)
the functionw(·,·) is real-analytic on (𝕋2)2=𝕋2×𝕋2 and has a unique nondegenerated minimum at (0,0)∈(𝕋2)2.The perturbationv=Φ*Φ is positive operator of rank one. Consequently, by well-known Weyl's theorem [12], the essential spectrum of Hμ(p) fills the following segment on the real axis:
(2.5)σess(Hμ(p))=σess(H0(p))=[m(p),M(p)],
where
(2.6)m(p)=min⁡q∈T2wp(q),M(p)=max⁡q∈T2wp(q).By Assumption2.1 there exist such δ-neighborhood Uδ(0)⊂𝕋2 of the point p=0∈𝕋2 and analytic vector function q0:Uδ(0)→𝕋2 that for any p∈Uδ(0) the point q0(p)=(q0(1)(p),q0(2)(p))∈𝕋2 is a unique nondegenerated minimum of the function wp(·) (see Lemma 3.2).Moreover, in the caseφ(q0(p))=0,p∈Uδ(0) the following integral
(2.7)∫T2φ2(s)dswp(s)-m(p)>0,
exists (see Lemma 3.4) and we introduce a parameter μ(p) as
(2.8)1μ(p)=∫T2φ2(s)dswp(s)-m(p)>0.
If φ(q0(p))≠0,p∈Uδ(0), then we define μ(p) as μ(p)=0.Definition 2.2.
The numberz=m(p) is called a virtual level of the operator Hμ(p), if the equation Hμ(p)f=m(p)f has a nonzero solution f∈L1(𝕋2)∖L2(𝕋2), where L1(𝕋2) is the Banach space of integrable functions on 𝕋2. The corresponding solution f is called a virtual state of the operator Hμ(p).In the following theorem we assert that for anyμ>μ(p) there exists a unique eigenvalue E(μ,p), lying below the essential spectrum, of the operator Hμ(p),p∈Uδ(0), but for 0<μ≤μ(p),p∈Uδ(0) the operator Hμ(p) has none eigenvalue outside the essential spectrum. It is proved that for fixed p∈Uδ(0), the function E(·,p) is analytic in (μ(p),+∞).Moreover, this theorem provides a criterion, for being the bottomm(p),p∈Uδ(0) of the essential spectrum of Hμ(p), a virtual level of the operator Hμ(p).Theorem 2.3.
Let Assumption2.1 holds and p∈Uδ(0). Then the following assertions are true.  (i)
Ifμ>μ(p), then the operator Hμ(p) has a unique eigenvalue E(μ,p), lying below the essential spectrum of Hμ(p). The function E(·,p) is monotonously decreasing real-analytic function in the interval (μ(p),+∞) and the function E(μ,·) is real-analytic in Uδ(0). The corresponding eigenfunction
(2.9)Ψ(μ;p,⋅,E(μ,p))=Cμφ(⋅)wp(⋅)-E(μ,p),
is analytic on 𝕋2, where C≠0 is a normalizing constant. Moreover, the mappings
(2.10)Ψμ:Uδ(0)⟶L2(T2),p⟼Ψ(μ;p,⋅,E(μ,p))∈L2(T2),Ψp:(μ(p),+∞)⟶L2(T2),μ⟼Ψ(μ;p,⋅,E(μ,p))∈L2(T2),
are vector-valued analytic functions in Uδ(0) and (μ(p),+∞), respectively.(ii)
Ifφ(q0(p))=0 and 0<μ<μ(p), then the operator Hμ(p) has none eigenvalue in (-∞,m(p)].(iii)
Ifφ(q0(p))=0, ∇φ(q0(p))=((∂φ/∂q1)(q0(p)),(∂φ/∂q2)(q0(p)))≠0 and μ=μ(p), then the number z=m(p) is a virtual level of the operator Hμ(p) and the corresponding virtual state is of the form:
(2.11)fp(⋅)=Cμ(p)φ(⋅)wp(⋅)-m(p)∈L1(T2)∖L2(T2),
where C≠0 is a normalizing constant.(iv)
Ifφ(q0(p))=0, ∇φ(q0(p))=((∂φ/∂q1)(q0(p)),(∂φ/∂q2)(q0(p)))=0 and μ=μ(p), then the number m(p)=wp(q0(p)) is an eigenvalue of the operator Hμ(p) and the corresponding eigenfunction is of the form
(2.12)fp(q)=Cμ(p)φ(⋅)wp(⋅)-m(p)∈L2(T2),
where C≠0 is a normalizing constant.Remark 2.4.
Notice that ifφ(q0(p))≠0, then μ(p)=0. So, in this case the number z=m(p) is neither a virtual level nor an eigenvalue for the operator Hμ(p).Remark 2.5.
From the positivity ofΦ*Φ it follows that the operator Hμ(p) has none eigenvalue lying above M(p).


## 3. Proof of the Results

We postpone the proof of the theorem after several lemmas and remarks.For anyμ>0 and p∈𝕋2 we define in ℂ∖[m(p);M(p)] an analytic function Δ(μ,p;·) (the Fredholm determinant Δ(μ,p;·), associated to the operator Hμ(p)) as
(3.1)Δ(μ,p;⋅)=1-μΩ(p;⋅),
where
(3.2)Ω(p;z)=∫T2φ2(s)dswp(s)-z,p∈T2,z∈C∖[m(p);M(p)].Lemma 3.1.
For anyμ∈(μ(p),+∞) and p∈Uδ(0) the number z∈ℂ∖σess(Hμ(p)),p∈𝕋2 is an eigenvalue of the operator Hμ(p) if and only if, when
(3.3)Δ(μ,p;z)=0.
The corresponding eigenfunction
(3.4)fμ,p(⋅)=Cμφ(⋅)wp(⋅)-z,
is analytic on 𝕋2, where C≠0 is a normalizing constant [4].Lemma 3.2.
Let Assumption2.1 holds. Then there exist such a δ-neighborhood Uδ(0)⊂𝕋2 of the point p=0 and analytic function q0:Uδ(0)→𝕋2 that for any p∈Uδ(0) the point q0(p) is a unique non degenerated minimum of wp(·).Proof.
By Assumption2.1, the square matrix
(3.5)A(0)=(∂2w0∂qi∂qj(0))i,j=12>0,
is positively defined and ∇w0(0)=0. Then by the implicit function theorem in analytic case there exist a δ-neighborhood Uδ(0)⊂𝕋2 of p=0∈𝕋2 and a unique analytic vector function q0(·):Uδ(0)→𝕋2 such that ∇wp(q0(p))=0 and
(3.6)A(p)=(∂2wp∂qi∂qj(q0(p)))i,j=12>0,p∈Uδ(0).
Hence for any p∈Uδ(0) the point q0(p) is a unique non degenerated minimum of the function wp(·).Remark 3.3.
We note that by the parametrical Morse lemma for anyp∈Uδ(0) there exists a map s=ψ(y,p) of the sphere Wγ(0)⊂ℝ2 with radius γ>0 and center at y=0 to a neighborhood U(q0(p)) of the point q0(p) that in U(q0(p)) the function wp(ψ(y,p)) can be represented as
(3.7)wp(ψ(y,p))=m(p)+y2.
Here the function ψ(y,·) (resp., ψ(·,p)) is analytic in Uδ(0) (resp., Wγ(0)) and ψ(0,p)=q0(p). Moreover, the Jacobian J(ψ(y,p)) of the mapping s=ψ(y,p) is analytic in Wγ(0) and positive, that is J(ψ(y,p))>0 for all p∈Uδ(0) and for all y∈Wγ(0).Lemma 3.4.
Let Assumption2.1 holds. Then the integral
(3.8)ξ(p)=∫T2φ2(s)-φ2(q0(p))wp(s)-m(p)ds,
exists and defines an analytic function in Uδ(0).Proof.
We represent the function(3.9)ξ(p,z)=∫T2φ2(s)-φ2(q0(p))wp(s)-zds,
as
(3.10)ξ(p,z)=ξ1(p,z)+ξ2(p,z),
where
(3.11)ξ1(p,z)=∫U(q0(p))φ2(s)-φ2(q0(p))wp(s)-zds,ξ2(p,z)=∫T2∖U(q0(p))φ2(s)-φ2(q0(p))wp(s)-zds,
and U(q0(p)) is a neighborhood of q0(p).
Observe that by Assumption2.1 for any p∈Uδ(0) the function ξ2(p,z) is analytic at the point z=m(p).
According to Remark3.3 in the integral for ξ1(p,z) a change of variables s=ψ(y,p) implies
(3.12)ξ1(p,z)=∫Wγ(0)φ2(ψ(y,p))-φ2(q0(p))y2+m(p)-zJ(ψ(y,p))dy,
where J(ψ(y,p)) is the Jacobian of the mapping ψ(y,p).
Passing to spherical coordinates asy=rν, we obtain
(3.13)ξ1(p,z)=∫0γrr2+m(p)-z{∫Ω2[φ2(ψ(rν,p))-φ2(q0(p))]J(ψ(rν,p))dν}dr,
where Ω2 is a unit sphere in ℝ2 and dν its element. Inner integral can be represented as
(3.14)∫Ω2[φ2(ψ(rν,p))-φ2(q0(p))]J(ψ(rν,p))dν=∑n=1∞‍τn(p)r2n,
where the Pizetti coefficients τn(p),n=1,2,… are analytic in Uδ(0) [13].
Thus we have that(3.15)ξ1(p,z)=∑n=1∞‍τn(p)∫0γr2n+1drr2+m(p)-z.
From (3.15) it follows that the following limit exists
(3.16)ξ1(p)=lim⁡z→m(p)-0ξ1(p,z)=lim⁡z→m(p)-0∑n=1∞‍τn(p)∫0γr2n+1drr2+m(p)-z=∑n=1∞‍γ2n2nτn(p),
and, consequently,
(3.17)ξ(p)=lim⁡z→m(p)-0ξ(p,z)=ξ1(p)+ξ2(p),
where ξ2(p)=ξ2(p,m(p)). Observe that the functions in the right hand side of (3.17) are analytic in p∈Uδ(0). So, the function ξ(p) is analytic in p∈Uδ(0).Proposition 3.5.
Forζ<0 the following equalities hold:
(3.18)In(ζ)=∫0δr2n+1drr2-ζ=-12ζnln⁡(-ζ)+I^n(ζ),n=0,1,2,…,
where ln(-ζ) is real for ζ<0 and I^n(ζ) is an analytic function in a neighborhood of the origin [14].In the following lemma we establish an expansion ofΔ(μ,p;z) in a half neighborhood (m(p)-ε,m(p)) of the point z=m(p).Lemma 3.6.
Assume Assumption2.1. Then for any μ>0, p∈Uδ(0) and sufficiently small m(p)-z>0 the function Δ(μ,p;·) can be represented as the following convergent series:
(3.19)Δ(μ,p;z)=1-μα0(p)ln⁡(m(p)-z)+μ2ln⁡(m(p)-z)∑n=1∞‍αn(p)(m(p)-z)n-μF(p,z),α0(p)=-12φ2(q0(p))J(q0(p)),F(p,z)=∑n=0∞‍cn(p)(m(p)-z)n,
where αn(p),cn(p),n=0,1,2,… are real numbers.Proof.
The functionΩ(p;·) can be written as a sum of the following functions:
(3.20)Ω(p;⋅)=I1(p,⋅)+I2(p,⋅)+I3(p,⋅),
where
(3.21)I1(p,z)=φ2(q0(p))∫U(q0(p))dswp(s)-z,I2(p,z)=φ2(q0(p))∫T2∖U(q0(p))dswp(s)-z,I3(p,z)=∫T2(φ2(s)-φ2(q0(p)))dswp(s)-z,
and U(q0(p)) is a neighborhood of the point q0(p),p∈Uδ(0).
Sinceminq∈𝕋2wp(q)=wp(q0(p)) for any p∈Uδ(0), the function I2(p,z) is analytic at z=m(p). According to Lemma 3.4 the function I3(p,m(p)) is analytic in Uδ(0).
A change of variabless=ψ(y,p) in the integral I1(p,z) yields
(3.22)I1(p,z)=φ2(q0(p))∫Wγ(0)J(ψ(y,p))dym(p)+y2-z.
Passing to spherical coordinates by y=rν we obtain
(3.23)I1(p,z)=φ2(q0(p))∫0γ∫Ω2J(ψ(rν,p))rdνdrm(p)+r2-z,
and hence
(3.24)I1(p,z)=φ2(q0(p))∫0γ(∫Ω2J(ψ(rν,p))dν)rdrm(p)+r2-z,
where Ω2 is unit sphere in ℝ2. Since
(3.25)∫Ω2J(ψ(rν,p))dν=∑n=0∞‍α~n(p)r2n,
where α~n(p), n=0,1,… are the Pizetti coefficients, we get
(3.26)I1(p,z)=φ2(q0(p))∑n=0∞‍α~n(p)∫0γr2n+1drm(p)+r2-z,
where α~0(p)=J(q0(p)). Using Proposition 3.5 we have
(3.27)∑n=0∞‍α~n(p)∫0γr2n+1drm(p)+r2-z=-12ln⁡(m(p)-z)∑n=0∞‍αn(p)(m(p)-z)n+Φ(p,z),
where Φ(p,z)=∑n=0∞βn(p)(m(p)-z)n and α^n(p)=(-1)nα~n(p). Using relations (3.27) and (3.21) and putting (3.26) in (3.20) we get required relation (3.19).Lemma 3.7.
Let Assumption2.1 hold. Then for any p∈Uδ(0) consider  (i)
ifφ(q0(p))=∇φ(q0(p))=0, then
(3.28)fp(q)=φ(⋅)wp(⋅)-m(p)∈L2(T2);(ii)
ifφ(q0(p))=0,∇φ(q0(p))≠0, then fp∈L1(𝕋2)∖L2(𝕋2).Proof.
We consider the following integral:(3.29)I(p)=∫T2F(s)ds(wp(s)-m(p))k,
where F(·) is a continuous function on 𝕋2 and k∈N. By Lemma 3.2 for any p∈Uδ(0) the function wp(·) has a unique non degenerated minimum at q=q0(p). Then there exist a neighborhood U(q0(p))⊂𝕋2 of the point q=q0(p) and positive number cp>0 that
(3.30)cp≤wp(q)-m(p),q∈T2∖U(q0(p)).
We represent the functionI(·) as a sum of two functions:
(3.31)I(⋅)=I1(⋅)+I2(⋅),
where
(3.32)I1(p)=∫U(q0(p))F(s)ds(wp(s)-m(p))k,I2(p)=∫T2∖U(q0(p))F(s)ds(wp(s)-m(p))k.
From (3.30) we get that I2(p)<∞. In the integral for I1(p) making a change of variables s:=ψ(y,p) one obtains
(3.33)I1(p)=∫Wγ(0)F(ψ(y,p))J(ψ(y,p))dyy2k,
where J(ψ(y,p)) is the Jacobian of the mapping s=ψ(y,p).
(i) LetF(s)=φ2(s), k=2. Then from (3.33) we get
(3.34)I1(p)=∫Wγ(0)φ2(ψ(y,p))J(ψ(y,p))dyy4.
Passing to spherical coordinates byy=rν we get
(3.35)I1(p)=∫0γ(∫Ω2φ2(ψ(rν,p))J(ψ(rν,p))dν)r-3dr.
Expanding the function φ(ψ(rν,p)) to the Taylor series at r=0 we obtain
(3.36)φ(ψ(rν,p))=φ(q0(p))+∑i=12‍∂φ∂ψ(i)(q0(p))(∑j=12‍∂ψ(i)∂yj(0,p)νj)r+g(r,ν)r2,yj=rνj,
where g(·,ν) is continuous in Wγ(0) and ν12+ν22=1. By condition of part  (i) of this lemma and from equality (3.36) it follows that (3.35) has the following form:
(3.37)I1(p)=∫0γG(p,r)dr,G(p,r)=r∫Ω2g2(r,ν)J(ψ(rν,p))dν.
Since the functionG(p,·) is continuous in [0,γ], we have I1(p)<∞. Taking into account ∥f∥L2(𝕋2)2=I(p), from (3.31) we get that f∈L2(𝕋2).
Now we show that if the conditions of part  (i) of Lemma3.7 are not satisfied, that is, φ(q0(p))≠0 or ∇φ(q0(p))≠0, then the function defined by (3.28) does not belong to L2(𝕋2).
Letφ(q0(p))=0 and ∇φ(q0(p))≠0. We will show that
(3.38)C(ν)=∑i=12‍∂φ∂ψ(i)(q0(p))(∑j=12‍∂ψ(i)∂yj(0,p)νj)≠0,ν∈Ω2.
Assume the converse, let
(3.39)∑i=12‍ci∑j=12‍uijνj=∑j=12‍∑i=12‍ciuij⋅νj=0,
where ci=(∂φ/∂ψ(i))(q0(p)) and uij=(∂ψ(i)/∂yj)(0,p),i,j=1,2. Since the function νj,j=1,2 are linearly independent, the last equality holds if and only if, when
(3.40)∑i=12‍ciuij=0,j=1,2.
Observe that det(uij)i,j=12=J(q0(p))≠0. Consequently, the equalities (3.40) hold if and only if, when c1=c2=0. This contradicts the fact that ∇φ(q0(p))≠0. Thus, C(ν)≠0. Hence the equality (3.35) has the form
(3.41)I1(p)=∫0γG~(p,r)dr,G~(p,r)=r-1∫Ω2g~2(r,ν)J(ψ(rν,p))dν,g~(r,ν)=C(ν)+g(r,ν)r.
Since
(3.42)∫0γr-1dr=∞,lim⁡r→0G~(p,r)r-1=J(q0(p))∫Ω2C2(ν)dν>0,
by the theorem on comparison of improper integrals, we get that I1(p)=∞ and thereforef∉L2(𝕋2).
In case ofφ(q0(p))≠0 the relation f∉L2(𝕋2) can be proven analogously.
(ii) LetF(s)=|φ(s)|,k=1. Then from (3.33) we get
(3.43)I1(p)=∫Wγ(0)|φ(ψ(y,p))|J(ψ(y,p))dyy2.
Passing to spherical coordinates by y=rν we obtain
(3.44)I1(p)=∫0γ[∫Ω2|φ(ψ(rν,p))|J(ψ(rν,p))dν]r-1dr.
By the condition of part  (ii) of Lemma 3.7 and from (3.36) we get
(3.45)I1(p)=∫0γ[∫Ω2|C(ν)+g(r,ν)r|J(ψ(rν,p))dν]dr.
Since the function under the integral sign is continuous in [0,γ], it follows that I1(p)<∞. Thus I(p)<∞. Taking into account ∥f∥L1(𝕋2)=I(p) we obtain f∈L1(𝕋2). Consequently, from part  (i) of Lemma 3.7 it follows that f∈L1(𝕋2)∖L2(𝕋2).Lemma 3.8.
Let the points=q0(p),p∈Uδ(0) a unique non degenerated minimum of the function wp(s), and φ(q0(p))=0. Then for any μ>0 the equation
(3.46)Hμ(p)f=m(p)f.
has a nonzero solution if and only if
(3.47)Δ(μ,p;m(p))=1-μ∫T2φ2(q)dqwp(q)-m(p)=0.
In this case the nonzero function
(3.48)fμ,p(⋅)=Cμφ(⋅)wp(⋅)-m(p)∈L1(T2),
is a solution of (3.46), where C≠0 a normalizing constant.This lemma can be proved as Lemma3.1 taking into account part (ii) of Lemma 3.7.Now we prove the main results.Proof of Theorem2.3.
(i) Observe thatΔ(μ,p;·) is continuous and monotonously decreasing in (-∞,m(p)). Moreover,
(3.49)lim⁡z→-∞Δ(μ,p;z)=1.
By definition, ifφ(q0(p))≠0, then μ(p)=0. So, for μ>μ(p)=0, Lemma 3.6 gives that
(3.50)lim⁡z→m(p)-0Δ(μ,p;z)=-∞.
Analogously, ifφ(q0(p))=0, then for μ>μ(p)>0 the inequality
(3.51)lim⁡z→m(p)-0Δ(μ,p;z)=1-μμ(p)<0,
holds.
The continuity of functionΔ(μ,p;·) in (-∞,m(p)) yields that the equation Δ(μ,p;z)=0 has a unique solution z=E(μ,p)<m(p) and hence Lemma 3.1 yields that the operator Hμ(p), p∈Uδ(0) has a unique eigenvalue E(μ,p).
Since for anyp∈Uδ(0) and μ∈(μ(p),+∞) the number z=E(μ,p) is a solution of the equation Δ(μ,p;z)=0 and the function Δ(μ,·;z) (resp., Δ(·,p;z)) is real analytic in Uδ(0) (resp., (μ(p),+∞)), the implicit function theorem implies that E(μ,·) (resp., E(·,p)) is real analytic in Uδ(0) (resp., (μ(p),+∞)).
Note that the functionΔ(·,p;z) monotonously decreases in (μ(p),∞) and hence for any μ1>μ2>μ(p) the eigenvalues E(μ1,p) and E(μ2,p) satisfy the relations:
(3.52)0=Δ(μ1,p;E(μ1,p))=Δ(μ2,p;E(μ2,p))>Δ(μ1,p;E(μ2,p)).
Using the monotonicity of the functionΔ(μ1,p;·) in (-∞,m(p)) we obtain that E(μ1,p)<E(μ2,p), that is, E(·,p) is monotonously decreases in (μ(p),∞).
Lemma3.1 implies that if for any μ∈(μ(p),+∞) and p∈Uδ(0) the number E(μ,p) is an eigenvalue of the operator Hμ(p), p∈Uδ(0), then the function
(3.53)Ψ(μ;p,⋅,E(μ,p))=Cμφ(⋅)wp(⋅)-E(μ,p)∈L2(T2),
is a solution of the equation
(3.54)Hμ(p)Ψ(μ;p,⋅,E(μ,p))=E(μ,p)Ψ(μ;p,⋅,E(μ,p)),
where C≠0 is a a normalizing constant.
Analyticity of the functionΨ(μ;p,·,E(μ,p)) follows from the analyticity of the functions φ(·) and [wp(·)-E(μ,p)]-1 in 𝕋2 and the representation (3.53). The functions E(μ,·) and w(·,q) are analytic in Uδ(0) and for any q∈𝕋d the inequality wp(q)-E(μ,p)>0 holds, therefore representation (3.53) yields that the mapping p↦Ψ(μ;p,·,E(μ,p)) is analytic in Uδ(0). Analogously the analyticity of the function E(·,p)) implies that the mapping μ↦Ψ(μ;p,·,E(μ,p)) is analytic in (μ(p),+∞).
(ii) Letφ(q0(p))=0 and 0<μ<μ(p). Since
(3.55)lim⁡z→m(p)-0Δ(μ,p;z)=Δ(μ,p;m(p))=1-μμ(p)>0,
we have Δ(μ,p;z)>0,z∈(-∞,m(p)] and Lemma 3.1 yields that the operator Hμ(p), p∈Uδ(0) does not have any eigenvalue in (-∞,m(p)].
The statements (iii) and (iv) of Theorem2.3 follows from Lemmas 3.7 and 3.8.



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*Source: 180953-2012-08-30.xml*