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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 875194, 12 pages
On the Discrete Spectrum of a Model Operator in Fermionic Fock Space
1Department of Mathematics, Faculty of Science, University Putra Malaysia, Malaysia
2Samarkand State University, Samarkand, Uzbekistan
3Institute For Mathematical Research, UPM, Malaysia
4Academic Lyceum Number One under Samarkand Institute of Economics and Service, Samarkand, Uzbekistan
Received 19 January 2013; Revised 15 April 2013; Accepted 21 April 2013
Academic Editor: Pavel Kurasov
Copyright © 2013 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.
We consider a model operator associated with a system describing three particles in interaction, without conservation of the number of particles. The operator acts in the direct sum of zero-, one-, and two-particle subspaces of the fermionic Fock space####^~^~^~^~^~^####x2009; over . We admit a general form for the "kinetic" part of the Hamiltonian , which contains a parameter to distinguish the two identical particles from the third one. (i) We find a critical value for the parameter that allows or forbids the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model has a threshold resonance) and we prove that only for the Efimov effect is absent, while this effect exists for any . (ii) In the case , we also establish the following asymptotics for the number of eigenvalues of below , for all .
In the spectral theory of the continuous and lattice three-particle Schr####^~^~^~^~^~^####xf6;dinger operators in , there is the remarkable phenomenon known as Efimov effect: if all Hamiltonians of the two-body subsystems are nonnegative and if at least two of them have a zero-energy resonance, then the three-body system has an infinite number of negative eigenvalues accumulating at zero.
This remarkable spectral property was discovered by Efimov  and has since become the subject of many papers [2####^~^~^~^~^~^####x2013;11]. The first mathematical proof of the existence of this effect was given by Jafaev , and Szlach####^~^~^~^~^~^####xe1;nyi and Vecserny####^~^~^~^~^~^####xe9;s  established the asymptotics for the number of eigenvalues near the threshold of the essential spectrum.
Recently, Wang  has proved the existence of the Efimov effect in the system with particles in but in this case the properties of the spectrum have not been fully comprehended yet.
In statistical physics [13, 14], solid-state physics [15, 16], and the theory of quantum fields [17####^~^~^~^~^~^####x2013;19], some important problems arise where the number of quasiparticles is bounded, but not fixed. The authers of  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 [13, 16####^~^~^~^~^~^####x2013;18, 20, 21].
The model operator, associated with a system describing two bosons and one particle, another nature in interaction, without conservation of the number of particles, was considered in [22, 23], and the existence of the Efimov effect was proved. This model operator acts in the direct sum of zero-, one-, and two-particle subspaces of the bosonic Fock space over .
In the present paper, we consider a model operator , acting in the direct sum of zero-, one-, and two-particle subspaces of the fermionic Fock space over , associated with a system describing two identical fermions and one particle, another nature in interactions, without conservation of the number of particles on the three-dimensional lattice.
The main aim of the present paper is to study spectral properties for a model operator with emphasis on the asymptotics for the number of infinitely many eigenvalues (Efimov's effect case).
We admit a general form for the kinetic part of the hamiltonian , which contains a parameter to distinguish the two identical particles from the third one (this parameter would be the ratio of the mass). Under some smoothness assumptions, we obtain the following results.(i)We find a critical value for the parameter that allows or forbids the Efimov effect and we prove that only for the Efimov effect is absent, while this effect exists for any . (ii)In the case , we also establish the following asymptotics for the number of eigenvalues below :
We notice that the assertion is surprising and similar assertions are not true for the lattice model operators in the bosonic Fock space [22, 23], but in both cases, the presence of the Efimov effect is due to the annihilation and creation operators.
However, the presence of the Efimov effect for the three-particle Schr####^~^~^~^~^~^####xf6;dinger operators is due to the two-particle interaction operators (see, e.g., [8####^~^~^~^~^~^####x2013;11, 24, 25] in lattice case).
We remark that for a model operator associated to a system describing three particles on the lattice the authors Dell'Antonio et al.  found an explicit value of the parameter , say , such that only for values of below this number, the Efimov effect is absent for the sector of the Hilbert space which contains functions which are antisymmetric with respect to the two identical particles, while it is present for all values of the parameter on the symmetric sector. In case of antisymmetric wavefunction, the number is a critical value for the mass ratio, where the Efimov effect is present or absent.
In the continuous case, interestingly the case of a system of three fermions, two identical and different from the third one, with short-range interaction, was also considered from a more physical point of view by Petrov , and he also found a critical value for the mass ratio ####^~^~^~^~^~^####x2248;13,6 (in our case ) that permits or forbids the Efimov effect.
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 Friedrichs models , , are studied, the location and structure of the essential spectrum of are given, and the main result of the paper is formulated. Section 3 deals with the review the Birman-Schwinger principle for the operator . In Section 4, we prove of the main result.
In order to facilitate a description of the content of this paper, we briefly introduce the notation used throughout this paper. Let be the three-dimensional torus (the first Brillouin zone, i.e., the dual group of ) equipped with its Haar measure. Denote by the Hilbert space of square-integrable functions defined on a measurable set . Denote by the subspace of antisymmetric functions of the Hilbert space . We denote by , several constants, whose magnitudes are not of interest.
2. The Model Operator and Statement of the Main Results
The Hilbert space is called the direct sum of zero-, one-, and two-particle subspaces of a fermionic Fock space over .
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 is an operator that increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator.
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, the model operator associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, acts in the Hilbert space as a matrix operator where the operators ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; are defined by the forms
Here , , -fixed real number, , -real-valued analytic functions on and is a real-analytic symmetric function defined on .
Under these assumptions, the operator is bounded and self-adjoint in .
Throughout this paper we assume the following additional technical assumptions.
Hypothesis 1. (a) The real-analytic function which is symmetric on is even with respect to and has a unique nondegenerate zero minimum at the origin and there exist positive definite matrix and positive real number such that (b) The functions and on are even and has a unique minimum at the origin.
By Hypothesis 1, for any the integral is finite and hence it defines a continuous even function on , which will be denoted by .
Since the function has a unique nondegenerate minimum at the point and the function is positive.
Hypothesis 2. For any nonzero , the inequality holds.
Remark 1. For the functions where is a real number and is real valued analytic conditionally negative definite function with a unique minimum at the origin, Hypotheses 1 and 2 are fulfilled (see [22, Lemma A1]).
Recall that a complex-valued bounded function is called conditionally negative definite if and for all and all satisfying .
2.1. The Friedrichs Model
To formulate the main results of the paper we introduce a family of Friedrichs models , , which act in with the entries where .
Let the operator , , act in as
The perturbation of the operator is a self-adjoint operator of rank 2. Therefore, in accordance with the invariance of the essential spectrum under finite rank perturbations, the essential spectrum of fills the following interval on the real axis: where
Definition 2. Let . The operator is said to have a threshold resonance if the number is an eigenvalue of the operator and the associated eigenfunction (up to a constant factor) satisfies the condition .
For any , we define an analytic function (the Fredholm determinant associated with the operator ) in by Since is continuous in , the following finite limit exists:
Lemma 4. For any the operator has an eigenvalue if and only if .
Proof. One can see that the equation
is equivalent to the following equation:
Putting to the first equation and then according to the equality we have the fact that (17) is equivalent to the equation
Thus (17) has nontrivial solution if and only if .
Lemma 5. (i) The operator has a threshold resonance if and only if and .(ii)If , then the operator has no a threshold resonance.(iii)Assume .(a)If , then the operator has a threshold resonance and the vector , where obeys the equation .(b)If , then the number is an eigenvalue of the operator and the vector , where and defined by (20), is the corresponding eigenvector.
Proof. (i) ####^~^~^~^~^~^####x201c;Only If Part.####^~^~^~^~^~^####x201d; Suppose that the operator has a threshold energy resonance. Then by Definition 2 the number is an eigenvalue of the operator (14) as
and the associated eigenfunction satisfies .
This solution is equal to the function (up to a constant factor) and hence .
####^~^~^~^~^~^####x201c;If Part.####^~^~^~^~^~^####x201d; Let the equality hold and let . Then the inequality holds and the function is an eigenfunction of ; that is, by Definition 2 the operator has a threshold energy resonance.(ii)Since and , we get which ends the proof together with .(iii)Similarly to the proof of Lemma 4, one can check that holds, and it is a determinant of the equation which is equivalent to the equation
Thus (23) has nontrivial solution if and only if .
By Hypothesis 1 there exist , and a bounded, partially continuous function such that (a)If by (i), the operator has a threshold resonance. Due to the fact and (24), we have .(b)If , then using (24) we can see and , with eigenfunction of corresponding to the eigenvalue .
Let be the matrix defined in Hypothesis 1 and stands for a -neighborhood of the origin. The following Lemma 6 plays a crucial role in the proof of the infiniteness (finiteness, resp.) of the number of eigenvalues lying below the bottom of the essential spectrum for a model operator . The proof may be handled in much the same way as in [22, 23].
Lemma 6. Let the operator have a threshold resonance. Then for any , sufficiently small and , the following decomposition holds, where (, resp.) is a function behaving like (, resp.) as ( uniformly in , resp.).
Lemma 7. Let the operator have a threshold resonance. Then there exist positive numbers , and such that
Proof. By the assumption (a) of Hypothesis 1, we have
Since has a nondegenerate minimum at the origin and the equality Holds, we have
Consequently, these asymptotics and Lemma 6 yield (26) for some positive numbers , .
By Hypotheses 1 and 2, we have and , , and hence
Then the inequality (27) follows from and the continuity of the function .
2.2. The Essential Spectrum of
Theorem 8. For the essential spectrum of the operator , the equality holds, where is the discrete spectrum of the operator , .
The following lemma describes the location of the essential spectrum of the operator .
Lemma 9. Let Hypothesis 1 be fulfilled and has a threshold resonance. Then where .
Proof. According to Lemma 4, the zero of is the eigenvalues of , so
By analyticity of , the subset of the essential spectrum of is described as a finite union of nonintersecting closed intervals.
The function is decreasing in , respectively, with , and hence there may exist its unique zero lying below, respectively, above the closed interval .
So, the set may be described as the union of intervals and lying below and above of , and
By virtue of Hypothesis 2, we have and using Lemma 5 we get , and so , that is since is decreasing in . Then according to Lemma 4 we get .
Consequently, and Theorem 8 ends the proof of the lemma.
2.3. Statement of the Main Results
Henceforth we assume that has a threshold resonance.
According to Lemma 9 for the bottom of the essential spectrum the equality is valid, and by we denote the number of eigenvalues and counted according to their multiplicities of lying below .
Let be a solution of the equation
Since the function at the r.h.s. of (38) is continuous, strictly increasing and surjective from to , the number is a unique positive solution.
In the following main theorem, we aprecisely describe the dependence of the number of eigenvalues of on the parameters .
Theorem 10. Let Hypothesis 1 be fulfilled and let have a threshold resonance at . Then we have the following. (i)For any , the operator has a finite number of eigenvalues lying below the bottom of the essential spectrum.(ii)For any , the operator has infinitely many eigenvalues lying below . The function obeys the relation
It follows from the positivity of the limit (39) that the discrete spectrum of the operator is infinite.
Remark 11. The constant is given as a positive function depending only on the variable , .
3. The Birman-Schwinger Principle
For a bounded self-adjoint operator , we define by is equal to infinity if is in the essential spectrum of and if is finite, it is equal to the number of the eigenvalues of larger than . By the definition of , we have
In our analysis of the spectrum of , the crucial role is played by the self-adjoint compact Faddeev-Newton-type integral operator , , in the space with the entries
Now we describe with the entries of in (3). We split into the sum of two operators where .
According to the last representation and (3), we take where is the resolvent of ,
One can verify that is the multiplication operator by the determinant . So, logically we may denote by the multiplication operator by .
Lemma 13. The operator is compact and continuous in and
where and is the inverse of .
For entries of matrix operator , we have
The correctness of is shown by the fact , that is, if and only if , , where .
For any , we define an operator acting in the Hilbert space by the form and prove the equality
Let , and then for , we get that is, .
Using the equality where , , and taking realized analogously, we get and so
Observe that where is acting in the Hilbert space by the equality
Employing (57), we show , that is, if and only if , .
Consequently, for any , we have
3.1. The Birman-Schwinger Principle at the Threshold
It should be noted that the operator can be defined as a bounded operator even for the point by
The next lemma can be proved analogously as Lemma 13.
Lemma 15. For any , , the inequality occurs.
4. The Sketch of the Proof of the Main Results
Applying the asymptotics for and using Lemma 6, we have where
4.1. The Infiniteness of the Discrete Spectrum of
In this subsection we will derive the asymptotics (39) for the number of eigenvalues of .
Let be the operator in defined by where the is the integral operator in with the kernel where is the characteristic function of the region and
Lemma 16. Let the conditions of Theorem 10 be fulfilled. The operator belongs to the Hilbert-Schmidt class and is continuous in .
Proof. Applying asymptotics (62) and (64), one can estimate the kernel of the operator , , by the square-integrable function Hence the operator belongs to the Hilbert-Schmidt class for all . In combination with the continuity of the kernel of the operator in , this gives the continuity of in .
Let be the restriction of the integral operator to the subspace . One verifies that the operator is unitarily equivalent to the integral operator acting in , where and , with the kernel where
The equivalence is given by the unitary dilation
Furthermore, we may replace by respectively, since the error will be a Hilbert-Schmidt operator continuous up to .
We have denoted by the characteristic function of the ball . By the replacement, we obtain the integral operator in with a kernel
By the dilation where is the unit sphere in , , , , one sees that the operator is unitarily equivalent to the integral operator with the kernel , , , where
For the completeness, we reproduce the following lemma, which has been proved in .
Lemma 17. Let , where is compact and continuous in . Assume that for some function , , the limit exists and is continuous in . Then the same limit exists for and
The following theorem is important for the proof of the asymptotics (39).
Theorem 18. Let the conditions of the part (ii) of Theorem 10 be fulfilled. The following equalities hold.
Proof. The coefficient in the r.h.s of the asymptotics (80) will be expressed by means of the self-adjoint integral operator , , in , whose kernel depends on the scalar product of the arguments and has the form
For , define
The function is very important for the proof of the existence of the Efimov effect. Denote .
Similarly to , we can derive that where is the multiplication operator by number in the subspace of the harmonics of degree and are Legendre polynomials. It follows from (83) and Lemma 19 that Lemma####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;19. ####^~^~^~^~^~^####x2009;The following assertions are true. (a). (b), , and , .(c), , and for any , there exists a positive number such that , .
Proof. For the completeness of the proof of this lemma we reproduce in Appendix A.
The positivity of follows from the fact that , if , which is proved in Lemma 19.
We remark that for all the number is finite and The difference of the operators and is compact (up to unitarily equivalence) and hence, taking into account that and Lemma 17, we obtain the limit (80).
4.2. The Finiteness of the Discrete Spectrum of
Lemma 20. Let and the hypothesis of part (i) of Theorem 10 be fulfilled. Then there exists a number depending on , such that
Proof. See Appendix B.
Remark 21. In the proof of Lemma 20, we can see that the main part of the operator is unitarily equivalent to the direct sum of multiplication operators in by the functions , , in Lemma 19. So is not compact operator.
By virtue of Lemma 20 the operator cannot have many eigenvalues larger than , and hence
A. Proof of Lemma####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;19
Case . For any , the can be written in the form
Since the integrand in (A.3) is positive, we obtain that the function is also positive and even as . The function is strictly decreased on for and hence the function strictly decreases on .
Let . Then we have
The function defined on is continuous and strictly increases.
Moreover, Thus the equation has a unique simple solution and (, resp.) for (, resp.).
Case . The function is calculated by It is easy to see that the inequalities hold.
Then using the inequalities (A.7), we obtain the following inequality: From here it follows that for all
Case . By Lemma 3.2 of , we get that for all and .
Since the function is increasing and the is decreasing, we have
for any and .
Thus for any and we have , and for any (for any and , resp.) we have (resp. ).
These facts complete the proof of the lemma.
B. Proof of Lemma 20
Since the operator is Hilbert-Schmidt (see Lemma 16), Weyl's Theorem implies that .
The space of all functions having support in is an invariant subspace of the operator .
Let be the restriction of the operator to the invariant subspace .
The operator is unitary equivalent to the integral operator acting in