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8ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
NUMBER AND LOCATION OF EIGENVALUES OF GENERALIZED FRIEDRICHS MODEL WITH FINITE RANK
PERTURBATIONS Dustova Sh.B.1, Rasulov T.H.2
1Dustova Shahlo Bakhtiyorovna - Assistant; 2Rasulov Tulkin Husenovich - Candidate of Physical and Mathematical Sciences,
Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: in the present paper we study a generalized Friedrichs model A with finite rank perturbations. This model (Hamiltonian) is associated with the operator energy of non-
conserved bounded number of particles on a d -dimensional lattice. The Fredholm
determinant corresponding to the operator A is constructed. We choose the finite system of
the bounded self-adjoint operators {Aa } such that the union of discrete spectrum of Aa
is coincide with the discrete spectrum of A . The number and location of the eigenvalues of A is found.
Keywords: generalized Friedrichs model, non-local potential, molecular-resonance model, essential spectrum, eigenvalue.
Operators known as Friedrichs operators [1] and generalized Friedrichs operators [2] appear in a series of problems in analysis, mathematical physics, and probability theory. The latter operators act in the Hilbert space
H := C 0L2(Td),
where Td is the d -dimensional torus, according to the rule
A :=
A)0 Ли Л Aoi A« "I Vk
V k=i y
Here
aWo = wofo, Aifi = a Jj(t)fi(t)dt,
(Afi )(x) = wi (x)fi (x), (Vkfi )(x) = p sin Xk J sin tkfi (t)dt, k = 1,2,...,d ,
Td
f = (f0,fi) e H , X = (xi,...,xd) GTd ; w0 is a constant, v(-) and
Wi(-) are real-valued continuous functions on Td, a, P> 0 are the coupling parameters.
It is easy to verify that under these assumptions the model operator A is bounded and self-adjoint in H . In modern mathematical physics the operator A)i is called annihilation
operator and Aqi is called creation operator.
We note that the character of the spectrum, the structure of the resolvent, the form of the eigenvectors for the discrete and continuous spectra, and the existence and completeness of the wave operators naturally related to the ordinary Friedrichs model, i.e. to a self-adjoint
operator of the form
+,
was completely or partly studied in many works, see, e.g., the pioneering work [1] and also [3] and [4], where M C Rd is a manifold and D is a function of two variables on the M . It was established in [1] that in the case where M = [—1,1] C R, Wi (x) = X , and P > 0 is small, the operator B up to finitely many eigenvalues has an absolutely continuous spectrum and that this operator in its absolutely continuous subspace is unitarily equivalent to the operator Bq such that
(Bof)(x) = w (x)f (x), f e L2 (M, dx).
Generalized Friedrichs model itself was introduced in [2], where its eigenvalues and "resonances" (i.e., the singularities of the analytic continuation of the resolvent) were studied. This model also considered in some other publications, among which we mention [5]. The threshold resonance, threshold and usual eigenvalues of A in the case P = 0 were discussed in many works, see e.g. [6-16]. The number and location of the eigenvalues of the generalized model with rank 3 perturbations were studied in [17-22] and used to define the number of closed bounded intervals and also to study the structure of the essential spectrum of a corresponding 3 X 3 operator matrices. More general case were studied in [23, 24]. In contrast above mentioned papers here the study of the discrete spectrum of a generalized Friedrichs model is reduces to the investigation of the discrete spectrum of the finite system of operators simpler that considered one.
To study the essential and discrete spectrum of A , we introduce the following operators:
'q q ^
H ^ H,
A0 : H ^ H, A0 :=
0 Aq Vq aii y
rA)0 A01 * 0 ,A01 A11 y
Ak :L2(T ) ^L2(T ), Ak := AQ - Vki k = 1,...,d.
It is clear that the perturbation A — AQ of the operator AQ is a self-adjoint operator of
rank d + 2 . Therefore, in accordance with the Weyl theorem about the invariance of the essential spectrum under the finite rank perturbations, the essential spectrum of the operator
A coincides with the essential spectrum of the operator AQ. It is evident that cess ( AQ ) = [m; M], where the numbers m and M are defined by
m := min w1 (x), M := max w1 (x).
xeTd x<aTd
This yields <Jess ( A) = [m; M] .
For k = 0,1,...,d we define an analytic function A £ (•) (the Fredholm determinant associated with the operator Ak ) in C \[m; M] by
2
. 2 f v (t)dt
Aq(z) := Wq — z — tf2 J w
Td w1(t) — z 2
Ak(z):= 1 — PJ (sint£) dt, k = 1,...,d.
Td W1(t) — z
A simple consequence of the Birman-Schwinger principle and the Fredholm theorem imply that for any k = 0,1,...,d the operator Afc has an eigenvalue z e C \ [m; M] if
and only if Ak (z) = 0 . Therefore,
adJSC(Ak) = {zeC\[m;M]: Ak(z) = 0}.
In the rest part of this paper we assume that the functions v(-) and W1 (•) are the even functions on each variable. For example, the functions d d w1(x) = Z (1 — cosxk), v(x) = n cos(kxk) k=1 k=1 satisfy such conditions.
The following theorem describes the relation between the discrete spectrum of the operators A and Afc , 0,1,...,d .
Theorem 1. The number z e C \[m; M ] is an eigenvalue of A if and only if z is
an eigenvalue one of the operators Ak , 0,1,...,d . From Theorem 1 it follows that d
adisc (A) = U &disc (Ak ) k=0
and hence
CTdsc (A) = {z e C \[m; M]: (1 Ak (z) = 0}.
k=0
Usually the function A(-) defined in C \ [m; M] by
d
A(z):= ( Ak (z)
k=0
is called the Fredholm determinant associated with the operator A .
The next result establishes the number and location of the eigenvalues of the operator
A.
Theorem 2. For all values of the coupling parameters (X, P > 0 the operator A has at most d + 2 discrete eigenvalues (counting with the multiplicities) such that d +1 of them are located on the l.h.s. of m and one of them is located on the r. h.s. of M .
Since the operators Ak , 0,i,...,d have the simple structure than A , Theorems i and
2 plays crucial role in the investigation of the location and structure of the essential and
discrete spectrum of the corresponding operator matrices in the truncated Fock space.
References
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ESSENTIAL AND DISCRETE SPECTRUM OF THE THREE-PARTICLE MODEL OPERATOR HAVING TENSOR SUM FORM Kurbonov G.G.1, Rasulov T.H.2
1Kurbonov Gulomjon Gafurovich - Assistant; 2Rasulov Tulkin Husenovich - Candidate of Physical and Mathematical Sciences,
Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: this paper is devoted to the spectral analysis of a model operator (Hamiltonian) H j, jj> 0 associated to a system of three quantum particles on a two-dimensional
lattice. The operator H„ can be represented as a tensor sum of two linear bounded self-
adjoint Friedrichs models hj . For all values of the parameter JJ> 0 the existence of the
unique eigenvalue of the operators hj and H j are shown. Using the spectrum of hj
the essential spectrum of H j is described. The location of the branches of the essential
spectrum of H j is identified.
Keywords: Hamiltonian, quantum particles, lattice, dispersion function, tensor sum, Friedrichs model, eigenvalue, essential spectrum.
In models of solid state physics [1,2] and also in lattice quantum field theory [3], one considers discrete Schroedinger operators, which are lattice analogs of the three-particle Schroedinger operator in the continuous space. One of the important problem in the spectral analysis of Schroedinger operators (in both cases) is to find whether the discrete spectrum is
finite or infinite set. In the present paper we consider the Hamiltonian H which is related
with the system of three quantum particles on a two dimensional lattice and describe its
spectrum. We remark that Hamiltonian H can be represented as a tensor sum of two
linear bounded self-adjoint Friedrichs models hj .
For the convenience of the reader, first we give some information about the spectrum of tensor sum of operators [4]. Tensor sum and tensor product of Hilbert space operators can be thought as an extension to infinite dimensional spaces of the traditional Kronecker sum
and Kronecker product of matrices on finite dimensional spaces. Let Hj and H2 be the Hilbert spaces and H be the tensor product product of Hj and H2, that is,
