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Advances in Mathematical Physics
Volume 2014 (2014), Article ID 943868, 10 pages
The Faddeev Equation and the Essential Spectrum of a Model Operator Associated with the Hamiltonian of a Nonconserved Number of Particles
1Department of Mathematics, Faculty of Science, University Putra Malaysia, 43400 Serdang, Malaysia
2Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Malaysia
3Academic Lyceum Number One Under Samarkand Institute of Economics and Service, 140104 Samarkand, Uzbekistan
Received 13 February 2014; Accepted 9 April 2014; Published 2 June 2014
Academic Editor: Pavel Kurasov
Copyright © 2014 Zahriddin Muminov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A model describing a truncated operator H (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 over is investigated. The location of the essential spectrum of the model operator H is described by means of the spectrum of the Friedreich model . Moreover, for the resolvent of H, the Faddeev type system of integral equations is obtained.
The main goal of the present paper is to give a thorough mathematical treatment of some properties for a model operator 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 over .
In statistical physics Minlos and Spohn , Malyshev and Minlos , solid-state physics Mattis , Mogilner  and the theory of quantum fields Friedrichs , Buhler et al.  some important problems arise where the number of quasiparticles is bounded, but not fixed. Sigal et al.  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 describing 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 the cut subspace of the Fock space, consisting of 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 or bosonic Fock space, in some cases (see, e.g., [9, 10]) over . We also refer to  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 operator , described by means of the spectrum of the Friedreich model , , 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. Section 1 is an introduction to the whole work. In Section 2, the model operator is described as a bounded self-adjoint operator in . Some spectral properties of the corresponding channel operator and Friedrichs models , , are given in Section 3. Section 4 deals with the review of the Faddeev type system of integral equations for the eigenfunction of operator . 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
Let be the three-dimensional torus, the cube with appropriately identified sides. We remark that the torus will always be considered as an abelian group with respect to the addition and multiplication by the real numbers regarded as operations on modulo . Denote by the subspace of antisymmetric functions of the Hilbert space .
We also introduce the following notations: and let , respectively, be an identity operator, respectively, an inner product in , .
The Hilbert space is called the direct sum of zero-, one-, and two-particle subspaces of fermionic Fock space over .
For we define the following operator:
Let be annihilation (creation) operators  defined in the Fock space for (). 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 that for all . So, a model describing a truncated operator acts in the Hilbert space as a matrix operator and given by where the operators , , and are multiplication by the functions , , and in , , and , respectively.
Here is a fixed number; are real-valued continuous functions on ; is a real-valued continuous symmetric function on .
Remark that under these conditions the operator 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 operator , we also consider a bounded self-adjoint operator acting in with form
This operator has a characteristic property of a channel operator (see, e.g., ) of three-particle discrete Schrödinger operator. Therefore, we call it a channel operator, corresponding to the model operator . Note that the channel operator has a simpler structure than the , and therefore the study of the spectral properties of plays an important role in future studies of the spectrum of .
Since commutes with the abelian group of multiplication operators , by the function : the decomposition of into , where , implies that the operator is decomposed into a direct von Neumann integral (see, e.g., [21, Theorem XIII.84]):
Here , is a Friedrichs model being bounded, self-adjoint, and acting in by the rule where and , , is a multiplication operator by the function :
Theorem 2. For the essential spectrum of the operator the equality, holds, where is the discrete spectrum of the operator , and
3.1. The Spectral Properties of the Friedrichs Model ,
Since , and then in accordance with the stability of the essential spectrum under finite rank perturbations, the essential spectrum of coincides with the spectrum of , and, namely, the equality, holds, where and are defined by
For any we define an analytic function (the Fredholm determinant associated with the operator ) in by that is, where , , is the resolvent of .
Lemma 3. For any the number is an eigenvalue of if and only if .
Proof. The equation
that is, the system of equations,
is equivalent to the system,
The solutions of (17) and (19) are connected with relations
Since the determinant of (19) is equal to , the equation, , has nontrivial solution if and only if .
Denote by , , (resp., , ) the number of eigenvalues, counted according their multiplicities of lying below (resp., above ).
Lemma 4. For any fixed , the following are true:(a)if and are linearly dependent (independent), then (b)
Proof. Simple calculation gives that the numbers
are simple eigenvalues of
Since , where is the subspace spanned by and , we have if and are linearly independent and if and are linearly dependent.
Then, using , for the operator , we obtain the following: (i)if and are linearly independent, then has two negative eigenvalues (with multiplicities) and a unique (simple) positive one;(ii)if and are linearly independent, then has only one (simple) negative and one (simple) positive eigenvalue.
According to , , and the minimax principle the number (resp., satisfies
Hence, using (i), (ii), and , we get proof.
Corollary 5. Let . Consider the following:(a)if and are linearly (dependent) independent, then the function can have no more than one zero (two zeros) in the interval ;(b) can have only one zero in the interval .
4. The Faddeev Type Equation
Remark 6. Note that for any the operators is the multiplication operator by the function respectively, is the integral operator with the kernel in .
One may check that the operator is a multiplication operator by the function in the space , where ; is the resolvent of .
4.1. Faddeev Type Integral Equation
Set , , , and .
Let the following matrix operators and , for any , act in the space by form where , , is multiplication by the function : and the operators , are defined by
Note that is Hilbert-Schmidt operators for any and so is in .
Lemma 7. The operator , , is bounded and invertible, and the inverse operator has form where , , is a multiplication operator by the function : Moreover, for any the operator is compact.
Proof. Due to the definition of this operator is multiplication by matrix
Clearly, is continuous matrix-valued function in , a fact making the boundedness of .
Since and , for , then .
Consequently, for any the matrix is invertible and which gives a fact that the inverse operator is multiplication by in .
Since every component of is compact so is , a fact that together boundedness of ends the proof.
Next assertion establishes connection between eigenvalues of the operators and .
Lemma 8. The number is an eigenvalue of the operator if and only if the number is an eigenvalue of .
Proof. Necessity. Let be an eigenvalue of with a corresponding eigenfunction ; that is, a system
has a nontrivial solution .
Whereas for the resolvent exists from the third equation of (39) for , we obtain where since .
Now we change (39) to the equivalent system of equations, using , , and . For this purpose, putting (40) for in (39), (41), we get that is, due to (32), (33), and (34)
The last fact and Lemma 7 conclude
The system (39) has nontrivial solution, so is the system (44), a fact implying that the number 1 is an eigenvalue of the operator .
Sufficiency. Let , , have an eigenvalue with corresponding eigenfunction . Thus (44) has nontrivial solution. Multiplying (44) with the operator from left side we get ; that is,
Set Then the last two equations of (45) give
It is easy to see that the vector function satisfies the equation ; that is, the number is an eigenvalue of .
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 spectrum of coincides with the set ; that is,
Proof. First Part. . This inclusion will be showed using the Weyl criterion (see [22, VII.12]) by constructing an appropriate sequence of trial functions.
Let . First we consider case and .
Then due to Theorem 2 and Lemma 3 there exists such that and the system of equations, has an infinite number solutions, and some solutions have form with the condition .
Then from (49) we obtain
Let , , be a characteristic function of a set and Lebesgue measure of .
A sequences of trial functions is chosen by where
Here the function is found from the orthogonality condition ; that is, since where the notation admits function-value either or .
There existence of follows from the following assertion, and they may be constructed by the method of induction.
Proposition 10. There exists orthonormal system , satisfying conditions
Proof. Let . Then we chose so that , where
Set . We continue this process. Assume that we have constructed . Then a function is chosen so that and it is orthogonal to
Set . Thus we have built orthonormal system , satisfying the conditions of the proposal.
Continuation of the Proof of Theorem 9. Assume . Then
Define an orthonormal system .
We estimate the norm of as
Note that is an orthogonal system, a fact that follows from disjointness of the supports of and .
From the equality, and (59) we take the uniformly boundedness of the system ; that is, for all .
Then, due to the compactness of , the limit , as , holds.
Now we estimate . Using the Schwartz inequality we receive
By the continuity of the matrix-valued function , the limit , as , is obtained.
Therefore, the sequence of orthonormal vector functions satisfies , as , a fact that means . Consequently, since is arbitrary.
In case , we chose as and we can construct an appropriate sequence of trial functions as where is a characteristic function of the set is Lebesgue measure of , and is sign function.
Second Part. . Denote by the resolvent form of , where , , is its matrix “entries,” it’s rows we denote by
Setting where , , is the resolvent of the operator , we obtain the resolvent equation
We write (67) as the form that is,
Defining the unitary operator , we can take and an equality
Then, putting the third equation of (69) into , we take where is an unknown vector operator and has the form
According to we write equation for as
The operator is a compact operator-valued function on and is invertible; even is real and either very negative or very positive. The analytic Fredholm theorem (see, e.g., in [22, Theorem ]) implies that there is a discrete set such that exists and is analytic in and meromorphic in with finite rank residues. Thus the function is analytic in with finite rank residues at the points of .
By analytic continuation, this holds for any . Thus, for any such , the operator has a bounded inverse. Therefore consists of isolated points and only the frontier points of are possible their limit points. Finally, since has finite rank residues at , we conclude that belongs to the discrete spectrum of , which completes the proof of Theorem 9.
Now we derive resolvent form (64) of the operator .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by University Putra Malaysia under research university grant scheme (FRGS), and Project code is 5524430. The authors are indebted to the anonymous referees for a number of constructive comments.
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