# The Faddeev Equation and the Essential Spectrum of a Model Operator Associated with the Hamiltonian of a Nonconserved Number of Particles

Authors: Zahriddin Muminov; Fudziah Ismail; Jamshid Rasulov
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

A model describing a truncated operatorH (truncated with respect to the number of particles) and acting in the direct sum of zero-, one-, and two-particle subspaces of fermionic Fock space FaL2T3 over L2(T3) is investigated. The location of the essential spectrum of the model operator H is described by means of the spectrum of the Friedreich model h(p);p∈T3. Moreover, for the resolvent of H, the Faddeev type system of integral equations is obtained.


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## Body

## 1. Introduction

The main goal of the present paper is to give a thorough mathematical treatment of some properties for a model operatorH with emphasis on the essential spectrum and its resolvent. This operator, associated to a system describing two identical fermions and one particle of another nature in interactions, without conservation of the number of particles on the three-dimensional lattice, acts in the direct sum of zero-, one-, and two-particle subspaces of fermionic Fock space Fa(L2(T3)) over L2(T3).In statistical physics Minlos and Spohn [1], Malyshev and Minlos [2], solid-state physics Mattis [3], Mogilner [4] and the theory of quantum fields Friedrichs [5], Buhler et al. [6] some important problems arise where the number of quasiparticles is bounded, but not fixed. Sigal et al. [7] have developed geometric and commutator techniques to find the location of the spectrum and to prove absence of singular continuous spectrum for Hamiltonians without conservation of the particle number.Notice that the study of systems describingn particles in interaction, without conservation of the number of particles, is reduced to the investigation of the spectral properties of self-adjoint operators acting in thecut subspaceH(n) of the Fock space, consisting of r≤n particles [1, 4–8].In the works [9–16], the location of essential spectra of model operators, associated with a system describing three particles, without conservation of the number of particles, was investigated. However, the corresponding operators act in the direct sum of zero-, one-, and two-particle subspaces of Fock space orbosonic Fock space, in some cases (see, e.g., [9, 10]) over L2(T3). We also refer to [17] for essential spectrum of discrete Schrödinger operators, associated to a system of two identical fermions and one particle of another nature in interactions.Recently, the authors in [18, 19] have proved that the fermionic Fock space case has some assertions being related to the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model (FM) has a threshold resonance). These results show that this effect does not hold even this FM has a threshold resonance while it holds in the (bosonic) Fock space case. This fact pushes us to study the resolvent and essential spectrum of the investigating operator.In the present paper, under some smoothness assumptions we obtain the location of the essential spectrum of the model operatorH, described by means of the spectrum of the Friedreich model h(p), p∈T3, and we derive the Faddeev type system of integral equations for the components of the resolvent of this model operator and find the form for resolvent.The organization of the present paper is as follows. Section1 is an introduction to the whole work. In Section 2, the model operator is described as a bounded self-adjoint operator H in H(3). Some spectral properties of the corresponding channel operator and Friedrichs models H(p), p∈T3, are given in Section 3. Section 4 deals with the review of the Faddeev type system of integral equations for the eigenfunction of operator H. In Section 5, we represent the main results (Theorems 9 and 11) and the sketch of their proofs.


## 2. Conventions and Definition of the Model Operator

LetT3 be the three-dimensional torus, the cube (-π,π]3 with appropriately identified sides. We remark that the torus T3 will always be considered as an abelian group with respect to the addition and multiplication by the real numbers regarded as operations on R3 modulo (2πZ)3. Denote by Las2((T3)2) the subspace of antisymmetric functions of the Hilbert space L2((T3)2).We also introduce the following notations:(1)H0=C,H1=L2(T3),H2=Las2((T3)2),
and let Ij, respectively, 〈·,·〉j be an identity operator, respectively, an inner product in Hj, j=0,1,2.The Hilbert spaceH(3)=H0⊕H1⊕H2 is called thedirect sum of zero-, one-, and two-particle subspaces of fermionic Fock space Fa(L2(T3)) over L2(T3).Forg∈L2(T3) we define the following operator:
(2)g:L2(T3)⟶C,g(f1)=〈f1,g〉1,f1∈L2(T3),g*:C⟶L2(T3),g*f0=g(·)f0,f0∈C,Lg,Lsg:L2((T3)2)⟶L2(T3),Lg=I1⊗g,Lsg=g⊗I1,Lg*,Lsg*:L2(T3)⟶L2((T3)2),Lg*=I1⊗g*,Lsg*=g*⊗I1.LetHij be annihilation (creation) operators [5] defined in the Fock space for i<j  (i>j). We note that, in physics, an annihilation operator is an operator that lowers the number of particles in a given state by one; a creation operator being an adjoint of the annihilation operator is an operator that increases the number of particles in a given state by one.In this paper, we consider the case, where the number of annihilations and creations of the particles of the considering system is equal to 1. It means thatHij≡0 for all |i-j|>1. So, a model describing a truncated operator H acts in the Hilbert space H(3) as a matrix operator and given by
(3)H=(H00H010H10H11H120H21H22),
where
(4)H01=a,H10=a*,H12=Lb,H11=H110,H21=12(Lb*-Lsb*),H22=H220-Lφ*Lφ-Lsφ*Lsφ,
the operators H00, H110, and H220 are multiplication by the functions u0, u(·), and E(·,·) in H0, H1, and H2, respectively.Hereu0 is a fixed number; a(·),b(·),u(·),φ(·) are real-valued continuous functions on T3; E(·,·) is a real-valued continuous symmetric function on (T3)2.Remark that under these conditions the operatorH is bounded and self-adjoint.


## 3. The Spectrum of the Channel Operator and of the Friedrichs Model

To study the essential spectrum, along with the operatorH, we also consider a bounded self-adjoint operator Hch acting in H^=L2(T3)⊕L2((T3)2) with form
(5)Hch=(H1112Lb12Lb*H220-Lφ*Lφ).This operator has a characteristic property of a channel operator (see, e.g., [20]) of three-particle discrete Schrödinger operator. Therefore, we call it a channel operator, corresponding to the model operator H. Note that the channel operator Hch has a simpler structure than the H, and therefore the study of the spectral properties of Hch plays an important role in future studies of the spectrum of H.SinceHch commutes with the abelian group of multiplication operators Uα, by the function α(·):
(6)Uα(f1(p)f2(p,q))=(α(p)f1(p)α(p)f2(p,q)),α∈L2(T3),
the decomposition of H^ into H^=∫p∈T3H(2)dp, where H(2)=H0⊕H1, implies that the operator Hch is decomposed into a direct von Neumann integral (see, e.g., [21, Theorem XIII.84]):
(7)Hch=∫T3‍⊕H(p)dp.HereH(p),p∈T3, is a Friedrichs model being bounded, self-adjoint, and acting in H(2) by the rule
(8)H(p)=H0(p)+V,
where
(9)H0(p)=(000h0(p)),V=(u(p)12b12b*-φ*φ),
and h0(p), p∈T3, is a multiplication operator by the function ep(·):=E(p,·):
(10)(h0(p)f1)(q)=ep(q)f1(q),f1∈H1.Remark 1.
The spectral properties of such type of Friedrichs models are studied in [9, 10].According to the theorem on the spectrum of decomposable operators (see, e.g., [21, Theorem XIII.85]) from (7), we obtain the following.Theorem 2.
For the essential spectrumσess(H) of the operator H the equality,
(11)σ(Hch)=∪p∈T3{σd(H(p))}∪[Emin⁡,Emax⁡],
holds, where σd(H(p)) is the discrete spectrum of the operator H(p),p∈T3, and
(12)Emin⁡=min⁡p,q∈T3E(p,q),Emax⁡=max⁡p,q∈T3E(p,q).


### 3.1. The Spectral Properties of the Friedrichs ModelH(p), p∈T3

Sincerank⁡V≤3, and then in accordance with the stability of the essential spectrum under finite rank perturbations, the essential spectrum σess(h(p)) of h(p) coincides with the spectrum of h0(p), and, namely, the equality,
(13)σess(H(p))=σess(h0(p))=[m(p),M(p)],p∈T3,
holds, where m(p) and M(p) are defined by
(14)m(p)=min⁡q∈T3ep(q),M(p)=max⁡q∈T3ep(q).For anyp∈T3 we define an analytic function Δ(p,z) (the Fredholm determinant associated with the operator H(p)) in C∖[m(p),M(p)] by
(15)Δ(p;z)=(I0-φr0(p,z)φ*)×(H00(p)-zI0-12br0(p,z)b*)f0-12(br0(p,z)φ*)2f0,f0≡1;
that is,
(16)Δ(p;z)=(1-∫T3‍φ2(s)dsE(p,s)-z)×(u(p)-z-12∫T3‍b2(s)dsE(p,s)-z)-12(∫T3‍b(s)φ(s)dsE(p,s)-z)2,
where r0(p;z), z∈C∖[m(p),M(p)], is the resolvent of h0(p),p∈T3.Lemma 3.
For anyp∈T3 the number z∈C∖[m(p),M(p)] is an eigenvalue of H(p) if and only if Δ(p,z)=0.Proof.
The equation(17)H(p)f=zf,f∈H(2),
that is, the system of equations,
(18)(u(p)-z)f0+12bf1=0,-12r0(p;z)b*f0+r0(p;z)φ*φf1=f1,hhhhhhhhhhihhhhhhhf=(f0,f1)∈H(2),
is equivalent to the system,
(19)(H00(p)-zI0-12br0(p,z)b*)f0+12br0(p,z)φ*α=0,-12φr0(p,z)b*f0+(I0-φr0(p,z)φ*)α=0,hhhhhhhhhhhhhhhhhf0,α∈C.
The solutions of (17) and (19) are connected with relations
(20)f0=f0,α=φf1,f1(q)=r0(p;z)(-12b(q)f0+φ(q)α).
Since the determinant of (19) is equal to Δ(p;z), the equation, H(p)f=zf,f∈H(2), has nontrivial solution if and only if Δ(p;z)=0.Denote byn-(H(p),z), z≤inf⁡σess(H(p)), (resp., n+(H(p),z), z≥sup⁡σess(H(p))) the number of eigenvalues, counted according their multiplicities of H(p) lying below z (resp., above z).Lemma 4.
For any fixedp∈T3, the following are true:(a)
ifφ(·) and b(·) are linearly dependent (independent), then
(21)n-(H(p),m(p))≤1(n-(H(p),m(p))≤2);(b)
(22)n+(H(p),M(p))≤1.Proof.
Simple calculation gives that the numbers(23)z1,2=-u(p)±u2(p)+4∥b∥22z1<0<z2
are simple eigenvalues of
(24)V′=(u(p)12b12b*0).
SinceRange(V)=C⊕〈b,φ〉, where 〈b,φ〉 is the subspace spanned by b and φ, we have dimRange(V)=3 if φ(·) and b(·) are linearly independent and dimRange(V)=2 if φ(·) and b(·) are linearly dependent.
Then, using-φ*φ≤0, σess(V′)={0} for the operator V, we obtain the following: (i)
ifφ(·) and b(·) are linearly independent, then V has two negative eigenvalues (with multiplicities) and a unique (simple) positive one;(ii)
ifφ(·) and b(·) are linearly independent, then V has only one (simple) negative and one (simple) positive eigenvalue.
According toH(p)≥m(p)+V, H(p)≤M(p)+V, and the minimax principle the number n-(H(p),m(p)) (resp., n+(H(p),M(p))) satisfies
(25)n-(H(p),m(p))≤n-(m(p)+V,m(p))(resp.,n+(H(p),M(p))≤n+(M(p)+V,M(p))).
Hence, using (i), (ii), andn-(m(p)+V,m(p))=n-(V,0), n+(M(p)+V,M(p))=n+(V,0) we get proof.Corollary 5.
Letp∈T3. Consider the following:(a)
ifφ(·) and b(·) are linearly (dependent) independent, then the function Δ(p,·) can have no more than one zero (two zeros) in the interval (-∞,m(p));(b)
Δ(p,·) can have only one zero in the interval (M(p),∞).


## 3.1. The Spectral Properties of the Friedrichs ModelH(p), p∈T3

Sincerank⁡V≤3, and then in accordance with the stability of the essential spectrum under finite rank perturbations, the essential spectrum σess(h(p)) of h(p) coincides with the spectrum of h0(p), and, namely, the equality,
(13)σess(H(p))=σess(h0(p))=[m(p),M(p)],p∈T3,
holds, where m(p) and M(p) are defined by
(14)m(p)=min⁡q∈T3ep(q),M(p)=max⁡q∈T3ep(q).For anyp∈T3 we define an analytic function Δ(p,z) (the Fredholm determinant associated with the operator H(p)) in C∖[m(p),M(p)] by
(15)Δ(p;z)=(I0-φr0(p,z)φ*)×(H00(p)-zI0-12br0(p,z)b*)f0-12(br0(p,z)φ*)2f0,f0≡1;
that is,
(16)Δ(p;z)=(1-∫T3‍φ2(s)dsE(p,s)-z)×(u(p)-z-12∫T3‍b2(s)dsE(p,s)-z)-12(∫T3‍b(s)φ(s)dsE(p,s)-z)2,
where r0(p;z), z∈C∖[m(p),M(p)], is the resolvent of h0(p),p∈T3.Lemma 3.
For anyp∈T3 the number z∈C∖[m(p),M(p)] is an eigenvalue of H(p) if and only if Δ(p,z)=0.Proof.
The equation(17)H(p)f=zf,f∈H(2),
that is, the system of equations,
(18)(u(p)-z)f0+12bf1=0,-12r0(p;z)b*f0+r0(p;z)φ*φf1=f1,hhhhhhhhhhihhhhhhhf=(f0,f1)∈H(2),
is equivalent to the system,
(19)(H00(p)-zI0-12br0(p,z)b*)f0+12br0(p,z)φ*α=0,-12φr0(p,z)b*f0+(I0-φr0(p,z)φ*)α=0,hhhhhhhhhhhhhhhhhf0,α∈C.
The solutions of (17) and (19) are connected with relations
(20)f0=f0,α=φf1,f1(q)=r0(p;z)(-12b(q)f0+φ(q)α).
Since the determinant of (19) is equal to Δ(p;z), the equation, H(p)f=zf,f∈H(2), has nontrivial solution if and only if Δ(p;z)=0.Denote byn-(H(p),z), z≤inf⁡σess(H(p)), (resp., n+(H(p),z), z≥sup⁡σess(H(p))) the number of eigenvalues, counted according their multiplicities of H(p) lying below z (resp., above z).Lemma 4.
For any fixedp∈T3, the following are true:(a)
ifφ(·) and b(·) are linearly dependent (independent), then
(21)n-(H(p),m(p))≤1(n-(H(p),m(p))≤2);(b)
(22)n+(H(p),M(p))≤1.Proof.
Simple calculation gives that the numbers(23)z1,2=-u(p)±u2(p)+4∥b∥22z1<0<z2
are simple eigenvalues of
(24)V′=(u(p)12b12b*0).
SinceRange(V)=C⊕〈b,φ〉, where 〈b,φ〉 is the subspace spanned by b and φ, we have dimRange(V)=3 if φ(·) and b(·) are linearly independent and dimRange(V)=2 if φ(·) and b(·) are linearly dependent.
Then, using-φ*φ≤0, σess(V′)={0} for the operator V, we obtain the following: (i)
ifφ(·) and b(·) are linearly independent, then V has two negative eigenvalues (with multiplicities) and a unique (simple) positive one;(ii)
ifφ(·) and b(·) are linearly independent, then V has only one (simple) negative and one (simple) positive eigenvalue.
According toH(p)≥m(p)+V, H(p)≤M(p)+V, and the minimax principle the number n-(H(p),m(p)) (resp., n+(H(p),M(p))) satisfies
(25)n-(H(p),m(p))≤n-(m(p)+V,m(p))(resp.,n+(H(p),M(p))≤n+(M(p)+V,M(p))).
Hence, using (i), (ii), andn-(m(p)+V,m(p))=n-(V,0), n+(M(p)+V,M(p))=n+(V,0) we get proof.Corollary 5.
Letp∈T3. Consider the following:(a)
ifφ(·) and b(·) are linearly (dependent) independent, then the function Δ(p,·) can have no more than one zero (two zeros) in the interval (-∞,m(p));(b)
Δ(p,·) can have only one zero in the interval (M(p),∞).


## 4. The Faddeev Type Equation

Set(26)Σ=[Emin⁡,Emax⁡]∪σtwo,
where
(27)σtwo={T3z:z∈R∖[m(p),M(p)]Δ(p;z)=0forsomep∈T3}.Clearly, according to Theorem2 and Lemma 3 the equalities Σ=σ(Hch) and σtwo=∪p∈T3σd(H(p)) hold.Remark 6.
Note that for anya,b∈L2(T3) the operators
(28)LaR220(z)Lb*(LsaR220(z)Lsb*)resp.,LaR220(z)Lsb*(LsaR220(z)Lb*)
is the multiplication operator by the function
(29)∫T3a(s)b(s)¯dsE(p,s)-z(∫T3a(s)b(s)¯dsE(s,p)-z),
respectively, is the integral operator with the kernel
(30)a(s)b(p)¯E(p,s)-z(a(s)b(q)¯E(s,p)-z),
in L2(T3).One may check that the operator(31)D(z)=I1-LφR220(z)Lφ*H110-zI1-12LbR220(z)Lb*-12LφR220(z)Lb*
is a multiplication operator by the function Δ(·;z) in the space L2(T3), where R220(z)=(H220-zI2)-1; z∈C∖[Emin⁡,Emax⁡] is the resolvent of H220.


### 4.1. Faddeev Type Integral Equation

SetH¯=H¯0⊕H¯1⊕H¯2, H¯0=H0, H¯1=H1,  and H¯2=H1.Let the following matrix operatorsA(z) and K(z), for any z∈C∖Σ, act in the space H¯ by form
(32)A(z)=(A00(z)000A11(z)A12(z)0A21(z)A22(z)),K(z)=(K00(z)K01(z)0K10(z)K11(z)K12(z)0K21(z)K22(z)),
where Aij(z):H¯j→H¯i, i,j=0,1,2, is multiplication by the function aij(p,z):
(33)a00(p,z)≡1,a11(p,z)=H11-zI1-12LbR220(z)Lb*,a12(p,z)=12LbR220(z)Lφ*,a21(p,z)=a12(p,z)¯=12LφR220(z)Lb*,a22(p,z)=I1-LφR220(z)Lφ*,
and the operators Kij(z):H¯j→H¯i, i,j=0,1,2 are defined by
(34)K00(z)=(z+1)I0-H00,K01(z)=-a,K10(z)=K01*(z),K11(z)=-12LbR220(z)Lsb*,K12(z)=12LbR220(z)Lsφ*,K21(z)=K12*(z),K22(z)=-LφR220(z)Lsφ*.Note thatKij(z) is Hilbert-Schmidt operators for any z∈C∖Σ and so is K(z) in H¯.Lemma 7.
The operatorA(z), z∈C∖Σ, is bounded and invertible, and the inverse operator A-1(z) has form
(35)A-1(z)=(B00(z)000B11(z)B12(z)0B21(z)B22(z)),
where Bij(z):H¯j→H¯i, i,j=0,1,2, is a multiplication operator by the function bij(p,z):
(36)b00(p,z)≡1,b11(p,z)=a22(p,z)Δ(p;z),b12(p,z)=-a12(p,z)Δ(p;z),b21(p,z)=-a21(p,z)Δ(p;z),b22(p,z)=a11(p,z)Δ(p;z).
Moreover, for any z∈C∖Σ the operator T^(z)=A-1(z)K(z) is compact.Proof.
Due to the definition ofA(z) this operator is multiplication by matrix
(37)A(p,z)=(a00(p,z)000a11(p,z)a12(p,z)0a12(p,z)a22(p,z)).
Clearly,A(p,z) is continuous matrix-valued function in T3, a fact making the boundedness of A(z).
Sincedet⁡(A(p,z))=Δ(p;z) and Δ(p;z)≠0, p∈T3 for z∉Σ, then det⁡(A(p,z))≠0.
Consequently, for anyp∈T3 the matrix A(p,z) is invertible and
(38)A-1(p,z)=(a00(p,z)000a22(p,z)Δ(p;z)-a12(p,z)Δ(p;z)0-a21(p,z)Δ(p;z)a11(p,z)Δ(p;z)),
which gives a fact that the inverse operator A-1(z) is multiplication by A-1(p,z) in H¯.
Since every component ofK(z) is compact so is K(z), a fact that together boundedness of A-1(z) ends the proof.Next assertion establishes connection between eigenvalues of the operatorsH and T^(z)=A-1(z)K(z).Lemma 8.
The numberz∈C∖Σ is an eigenvalue of the operator H if and only if the number λ=1 is an eigenvalue of T^(z).Proof.
Necessity. Let z∈C∖Σ be an eigenvalue of H with a corresponding eigenfunction f; that is, a system
(39)(H00-z)f0+H01f1=0,H10f0+(H11-z)f1+H12f2=0,H21f1+(H22-z)f2=0,
has a nontrivial solution f=(f0,f1,f2).
Whereas forz∉[Emin⁡,Emax⁡] the resolvent R220(z)=(H220-zI2) exists from the third equation of (39) for f2, we obtain
(40)f2=R220(z)(12(Lφ*-Lsφ*)c-12(Lb*-Lsb*)f1),
where
(41)c=2Lφf2,
since Lsφf2(q)=-Lsφf2(q).
Now we change (39) to the equivalent system of equations, using f0, f1, and c. For this purpose, putting (40) for f2 in (39), (41), we get
(42)f0=((z+1)I0-H00)f0-af1,(H11-zI1-12LbR220(z)Lb*)f1+12LbR220(z)Lφ*c=-a*f0-12LbR220(z)Lsb*f1+12LbR220(z)Lsφ*c,12LφR220(z)Lb*f1+(I1-LφR220(z)Lφ*)c=12LφR220(z)Lsb*f1-LφR220(z)Lsφ*c;
that is, due to (32), (33), and (34)
(43)A(z)ψ=K(z)ψ,ψ=(f0,f1,c)∈H¯.
The last fact and Lemma7 conclude
(44)ψ=A-1(z)K(z)ψ,i.e.,ψ=T^(z)ψ.
The system (39) has nontrivial solution, so is the system (44), a fact implying that the number 1 is an eigenvalue of the operator T^(z).
Sufficiency. Let T^(z), z∈C∖Σ, have an eigenvalue λ=1 with corresponding eigenfunction ψ=(f0,f1,c)∈H¯. Thus (44) has nontrivial solution. Multiplying (44) with the operator A(z) from left side we get A(z)ψ=K(x)ψ; that is,
(45)(H00-zI0)f0+af1=0,a*f0+(H11-zI1)f1+Wf1+Lb(R220(z)(12(Lφ*-Lφ*)c-12(Lb*-Lsb*)f1))=0,c=Lφ(R220(z)(12(Lφ*-Lφ*)c-12(Lb*-Lsb*)f1)).
Set(46)f2=R220(z)(12(Lφ*-Lsφ*)c-12(Lb*-Lsb*)f1).
Then the last two equations of (45) give
(47)c=2Lφf2.
It is easy to see that the vector functionf=(f0,f1,f2)∈H satisfies the equation Hf=zf; that is, the number z is an eigenvalue of H.


## 4.1. Faddeev Type Integral Equation

SetH¯=H¯0⊕H¯1⊕H¯2, H¯0=H0, H¯1=H1,  and H¯2=H1.Let the following matrix operatorsA(z) and K(z), for any z∈C∖Σ, act in the space H¯ by form
(32)A(z)=(A00(z)000A11(z)A12(z)0A21(z)A22(z)),K(z)=(K00(z)K01(z)0K10(z)K11(z)K12(z)0K21(z)K22(z)),
where Aij(z):H¯j→H¯i, i,j=0,1,2, is multiplication by the function aij(p,z):
(33)a00(p,z)≡1,a11(p,z)=H11-zI1-12LbR220(z)Lb*,a12(p,z)=12LbR220(z)Lφ*,a21(p,z)=a12(p,z)¯=12LφR220(z)Lb*,a22(p,z)=I1-LφR220(z)Lφ*,
and the operators Kij(z):H¯j→H¯i, i,j=0,1,2 are defined by
(34)K00(z)=(z+1)I0-H00,K01(z)=-a,K10(z)=K01*(z),K11(z)=-12LbR220(z)Lsb*,K12(z)=12LbR220(z)Lsφ*,K21(z)=K12*(z),K22(z)=-LφR220(z)Lsφ*.Note thatKij(z) is Hilbert-Schmidt operators for any z∈C∖Σ and so is K(z) in H¯.Lemma 7.
The operatorA(z), z∈C∖Σ, is bounded and invertible, and the inverse operator A-1(z) has form
(35)A-1(z)=(B00(z)000B11(z)B12(z)0B21(z)B22(z)),
where Bij(z):H¯j→H¯i, i,j=0,1,2, is a multiplication operator by the function bij(p,z):
(36)b00(p,z)≡1,b11(p,z)=a22(p,z)Δ(p;z),b12(p,z)=-a12(p,z)Δ(p;z),b21(p,z)=-a21(p,z)Δ(p;z),b22(p,z)=a11(p,z)Δ(p;z).
Moreover, for any z∈C∖Σ the operator T^(z)=A-1(z)K(z) is compact.Proof.
Due to the definition ofA(z) this operator is multiplication by matrix
(37)A(p,z)=(a00(p,z)000a11(p,z)a12(p,z)0a12(p,z)a22(p,z)).
Clearly,A(p,z) is continuous matrix-valued function in T3, a fact making the boundedness of A(z).
Sincedet⁡(A(p,z))=Δ(p;z) and Δ(p;z)≠0, p∈T3 for z∉Σ, then det⁡(A(p,z))≠0.
Consequently, for anyp∈T3 the matrix A(p,z) is invertible and
(38)A-1(p,z)=(a00(p,z)000a22(p,z)Δ(p;z)-a12(p,z)Δ(p;z)0-a21(p,z)Δ(p;z)a11(p,z)Δ(p;z)),
which gives a fact that the inverse operator A-1(z) is multiplication by A-1(p,z) in H¯.
Since every component ofK(z) is compact so is K(z), a fact that together boundedness of A-1(z) ends the proof.Next assertion establishes connection between eigenvalues of the operatorsH and T^(z)=A-1(z)K(z).Lemma 8.
The numberz∈C∖Σ is an eigenvalue of the operator H if and only if the number λ=1 is an eigenvalue of T^(z).Proof.
Necessity. Let z∈C∖Σ be an eigenvalue of H with a corresponding eigenfunction f; that is, a system
(39)(H00-z)f0+H01f1=0,H10f0+(H11-z)f1+H12f2=0,H21f1+(H22-z)f2=0,
has a nontrivial solution f=(f0,f1,f2).
Whereas forz∉[Emin⁡,Emax⁡] the resolvent R220(z)=(H220-zI2) exists from the third equation of (39) for f2, we obtain
(40)f2=R220(z)(12(Lφ*-Lsφ*)c-12(Lb*-Lsb*)f1),
where
(41)c=2Lφf2,
since Lsφf2(q)=-Lsφf2(q).
Now we change (39) to the equivalent system of equations, using f0, f1, and c. For this purpose, putting (40) for f2 in (39), (41), we get
(42)f0=((z+1)I0-H00)f0-af1,(H11-zI1-12LbR220(z)Lb*)f1+12LbR220(z)Lφ*c=-a*f0-12LbR220(z)Lsb*f1+12LbR220(z)Lsφ*c,12LφR220(z)Lb*f1+(I1-LφR220(z)Lφ*)c=12LφR220(z)Lsb*f1-LφR220(z)Lsφ*c;
that is, due to (32), (33), and (34)
(43)A(z)ψ=K(z)ψ,ψ=(f0,f1,c)∈H¯.
The last fact and Lemma7 conclude
(44)ψ=A-1(z)K(z)ψ,i.e.,ψ=T^(z)ψ.
The system (39) has nontrivial solution, so is the system (44), a fact implying that the number 1 is an eigenvalue of the operator T^(z).
Sufficiency. Let T^(z), z∈C∖Σ, have an eigenvalue λ=1 with corresponding eigenfunction ψ=(f0,f1,c)∈H¯. Thus (44) has nontrivial solution. Multiplying (44) with the operator A(z) from left side we get A(z)ψ=K(x)ψ; that is,
(45)(H00-zI0)f0+af1=0,a*f0+(H11-zI1)f1+Wf1+Lb(R220(z)(12(Lφ*-Lφ*)c-12(Lb*-Lsb*)f1))=0,c=Lφ(R220(z)(12(Lφ*-Lφ*)c-12(Lb*-Lsb*)f1)).
Set(46)f2=R220(z)(12(Lφ*-Lsφ*)c-12(Lb*-Lsb*)f1).
Then the last two equations of (45) give
(47)c=2Lφf2.
It is easy to see that the vector functionf=(f0,f1,f2)∈H satisfies the equation Hf=zf; that is, the number z is an eigenvalue of H.


## 5. Formulation and Proof of the Main Results

The first main result of the paper is given in the following theorem.Theorem 9.
The essential spectrumσess(H) of H coincides with the set Σ; that is,
(48)σess(H)=Σ.Proof.
First Part.
σ
two
⊂
σ
ess
(
H
). This inclusion will be showed using the Weyl criterion (see [22, VII.12]) by constructing an appropriate sequence of trial functions.
Letz0∈Σ. First we consider case z0∈σtwo and z0∉[Emin⁡,Emax⁡].
Then due to Theorem2 and Lemma 3 there exists p0∈T3 such that det⁡(A(p0,z0))=Δ(p0,z0)=0 and the system of equations,
(49)α0=0,(H00-z0I0-12br0(p0,z0)b*)α1+12br0(p0,z0)φ*α2=0,12φr0(p0,z0)b*α1+(I0-φr0(p0,z0)φ*)α2=0,
has an infinite number solutions, and some solutions have form l=(0,α1,α2) with the condition α12+α22>0.
Then from (49) we obtain
(50)A(p0,z0)l=0,l=(0,α1,α2)∈C3.
Letχn(p), n∈N, be a characteristic function of a set
(51)Vn={p∈T3:1n+1<|p-p0|<1n},
and μ(Vn) Lebesgue measure of Vn.
A sequences of trial functions{f(n)} is chosen by
(52)f(n)=(0,f1(n),f2(n)),
where
(53)f1(n)=α1ψ(n),f2(n)=R220(z0)(α2(Lφ*-Lsφ*)ψ(n)-α12(Lb*-Lsb*)ψ(n)),ψ(n)=cnχnμ(Vn).
Here the functioncn∈L2(T3) is found from the orthogonality condition {f(n)}; that is,
(54)〈f(n),f(m)〉H(3)=-〈R220(z0)(α2Lφ*-α12Lb*)ψ(n),R220(z0)(α2Lsφ*-α12Lsb*)ψ(m)〉2=0,n≠m,
since
(55)〈R220(z0)Lα*ψ(n),R220(z0)Lβ*ψ(m)〉2=0,〈R220(z0)Lsα*ψ(n),R220(z0)Lsβ*ψ(m)〉2=0,hhhhhhhhhhhhhhhhhhhhhhhhihn≠m,
where the notation α,β admits function-value either b or φ.
There existence ofcn follows from the following assertion, and they may be constructed by the method of induction.Proposition 10.
There exists orthonormal system{cn}⊂L2(T3), satisfying conditions
(56)(a)supp⁡cn⊂Vn,(b)〈R220(z0)(α2Lφ*-α12Lb*)ψ(n),R220(z0)(α2Lsφ*-α12Lsb*)ψ(m)〉2=0.
Proof. Let c1=χ1/μ(V1). Then we chose c~2∈L2(V2) so that ∥c~2∥=1  (c~2,ε1(2))=0, where
(57)ε1(2)(p)=∫T3‍([α2φ(q)-(α1/2)b(q)][(α1/2)b(p)-α2φ(p)](E(p,q)-z0)2)×c1(q)χ2(q)dq.
Setc2(p)=c~2(p)χ2(p). We continue this process. Assume that we have constructed c1(p),…,cn(p). Then a function c~n+1∈L2(Vn+1) is chosen so that ∥c~n+1∥=1 and it is orthogonal to
(58)εi(n+1)(p)=∫T3‍([α2φ(q)-(1/2)α1b(q)][α1b(p)-2α2φ(p)](E(p,q)-z0)2)×ci(q)χn+1(q)dq,i=1,n¯.
Setcn+1(p)=c~n+1(p)χn+1(p). Thus we have built orthonormal system {cn}, satisfying the conditions of the proposal.Continuation of the Proof of Theorem9. Assume Δ(p0,z0)=0. Then
(59)∥f(n)∥2≥ξ02μ(Vn),ξ0=α12+α22∥φ2∥2(Emax⁡-z0)2.Define an orthonormal systemf~(n)=f(n)/∥f(n)∥.We estimate the norm of(H-z0)f~(n) as
(60)∥(H-z0)f~(n)∥2≤∥A(z0)ψ~(n)∥H¯2+∥K(z0)ψ~(n)∥H¯2,ψ~(n)=(0,ψ1(n)∥f(n)∥,ψ2(n)∥f(n)∥).Note that{ψ~(n)}⊂H¯ is an orthogonal system, a fact that follows from disjointness of the supports of ψ~(n) and ψ~(m).From the equality,(61)∥ψ~(n)∥2=1∥f(n)∥21μ(Vn)(α12+α22),
and (59) we take the uniformly boundedness of the system {ψ~(n)}; that is, ∥ψ~(n)∥2≤(1/ξ0)∥l∥C32 for all n∈N.Then, due to the compactness ofK(z0), the limit ∥K(z0)ψ~(n)∥→0, as n→∞, holds.Now we estimate∥A(z0)ψ~(n)∥. Using the Schwartz inequality we receive
(62)∥A(z0)ψ~(n)∥2≤C2sup⁡p∈Vn∥A(p,z0)l∥C32,whereC2=max⁡{2ξ0,8∥φ∥2ξ0}.By the continuity of the matrix-valued functionF(p)=A(p,z0), the limit sup⁡p∈Vn∥A(p,z0)l∥C3→0, as n→∞, is obtained.Therefore, the sequence of orthonormal vector functions{f~(n)} satisfies ∥(H-z0)f~(n)∥→0, as n→∞, a fact that means z0∈σess(H). Consequently, σtwo⊂σess(H) since z0 is arbitrary.In casez0∈[Emin⁡,Emax⁡], we chose (p0,q0)∈(T3)2 as z=E(p0,q0) and we can construct an appropriate sequence of trial functions as
(63)f(n)=(0,0,f2(n))T,f2(n)(p,q)=sgn⁡(p2-q2)χn(p,q)μ(Un),
where χn(p,q) is a characteristic function of the set Un=Vn×Vn⊂(T3)2,μ(Un) is Lebesgue measure of Un, and sgn⁡ is sign function.Consequently,[Emin⁡,Emax⁡]∪σtwo⊂σess(H).Second Part.
σ
ess
(
H
)
⊂
Σ. Denote by
(64)R(z)=(R00(z)R01(z)R02(z)R10(z)R11(z)R12(z)R20(z)R21(z)R22(z)),z∈Σ,
the resolvent form of H, where Rij(z):Hj→Hi, i·j=1,2,3, is its matrix “entries,” it’s rows Ri(z):H(3)→Hi,i=1,2,3 we denote by
(65)Ri(z)=(Ri0(z),Ri1(z),Ri2(z)).Setting(66)R0(z)=diag⁡{I0,I1,R220(z)},H0(z)=diag⁡{I0,I1,H220-zI2},
where R220(z)=(H220-zI2)-1, z∈C∖[Emin⁡,Emax⁡], is the resolvent of the operator H220, we obtain the resolvent equation
(67)R(z)=R0(z)-R0(z)((H-zI)-H0(z))R(z).In our case resolvent Equation (67) is not compact. Therefore to overcome this difficulty, we derive Faddeev type system of integral equations (see, e.g., [23]) for the components of the resolvent.We write (67) as the form
(68){R0(z)R1(z)R2(z))=R0(z)-R0(z)×(H00-(z+1)I0H010H10H11-(z+1)I1H120H21(H22-H220))×(R0(z)R1(z)R2(z));
that is,
(69)(R0(z)R1(z)R2(z))=R0(z)-((H00-zI0-I0)R0(z)+H01R1(z)H10R0(z)+(H11-zI1-I1)R1(z)+H12R2(z)R220(z)H21R1(z)-R220(z)(Lφ*Lφ+Lsφ*Lsφ)R2(z)).Defining the unitary operatorP:Las2((T3)2)→Las2((T3)2):Pf(p,q)=-f(q,p), we can take PLsφ*Lsφ=-Lφ*Lφ and an equality
(70)Lsφ*LsφR2(z)=-Lsφ*PLφR2(z).Then, putting the third equation(71)R2(z)=R220(z)-(R220(z)H21R1(z)-R220(z)(Lφ*-Lsφ*P)LφR2(z)),
of (69) into H12R2(z), we take
(72)R(z)=R0(z)-Q0(z)-H1X(z),
where
(73)Q0(z)=diag⁡{0,H12R220(z),0},X(z)=(R0(z),R1(z),LφR2(z))T,
is an unknown vector operator and H1:H¯→H¯ has the form(74)H1=(H00-(z+1)I0H010H10H11-(z+1)I1-H12R220(z)H21H12R220(z)(Lφ*-Lsφ*P)0R220(z)H21-R220(z)(Lφ*-Lsφ*P)).Set(75)L(z)=diag⁡{I0,I1,Lφ}.According toX(z)=L(z)R(z) we write equation for X(z) as(76)X(z)=L(z)R0(z)-L(z)Q0(z)-(H00-(z+1)I0H010H10H11-(z+1)I1-H12R220(z)H21H12R220(z)(Lφ*-Lsφ*P)0LφR220(z)H21-LφR220(z)(Lφ*-Lsφ*P))X(z).Due to (33), (34) we have
(77)H00-(z+1)I0=-I0-A00(z)-B00(z),H11-(z+1)I1-H12R220(z)H21=-I1-A11(z)-K11(z),-H12R220(z)(Lφ*-Lsφ*P)=A12(z)-K12(z)LφR220(z)H21=A21(z)-K21(z),LφR220(z)(Lφ*-Lsφ*P)=-I2+A22(z)-K22(z).Using these and (33), (34) we get
(78)(A(z)-K(z))X(z)=L(z)(R0(z)-Q0(z)).By virtue of Lemma7 the operator A(z), z∈C∖Σ, is invertible and then we get the Faddeev type equation [23]
(79)(I-A-1(z)K(z))X(z)=A-1(z)L(z)(R0(z)-Q0(z)),
where I is the identity operator on H¯.The operatorT^(z)=A-1(z)K(z) is a compact operator-valued function on C∖Σ and F(z)=I-T^(z) is invertible; even z is real and either very negative or very positive. The analytic Fredholm theorem (see, e.g., in [22, Theorem VI⁡.14]) implies that there is a discrete set S⊂C∖Σ such that F(z)-1 exists and is analytic in C∖(Σ∪S) and meromorphic in C∖Σ with finite rank residues. Thus the function (I-T^(z))-1A-1(z)L(z)(R0(z)-Q0(z))≡G(z) is analytic in C∖(Σ∪S) with finite rank residues at the points of S.Letz∉S and Im⁡z≠0; then, by (81) and (79) we have G(z)=X(z). In particular,
(80)R(z)(H-zI)=(R0(z)-Q0(z)-H1X(z))(H-zI)=I,
where I is the identity operator in H(3).By analytic continuation, this holds for anyz∉Σ∪S. Thus, for any such z, the operator H-zI has a bounded inverse. Therefore σ(H)∖Σ consists of isolated points and only the frontier points of Σ are possible their limit points. Finally, since R(z) has finite rank residues at z∈S, we conclude that σ(H)∖Σ belongs to the discrete spectrum σd(H) of H, which completes the proof of Theorem 9.Now we derive resolvent form (64) of the operator H.Theorem 11.
Letz∈C∖σ(H). Then the resolvent R(z)=(H-zI)-1 of H has the form
(81)R(z)=R0(z)-Q0(z)-H1(A(z)-K(z))-1L(z)(R0(z)-Q0(z)),
where A(z), K(z), R0(z), H1, and L(z) are defined by (33), (34), (66), (74), and (75), respectively.Proof.
Lemma8, (69), and (79) give the proof.



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*Source: 943868-2014-06-02.xml*