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1522. Muminov M.I., Rasulov T.H. Infiniteness of the number of eigenvalues embedded in the essential spectrum of a 2x2 operator matrix // Eurasian Mathematical Journal. 5:2 (2014). P. 60-77.
23. Rasulov T.Kh. Investigation of the spectrum of a model operator in a Fock space // Theoret. and Math. Phys. 161:2 (2009). P. 1460-1470.
24. Muminov M.I., Rasulov T.H. The Faddeev equation and essential spectrum of a Hamiltonian in Fock Space // Methods Funct. Anal. Topology. 17:1 (2011). P. 47-57.
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^, 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 h^ . For all values of the parameter 0 the existence of the
unique eigenvalue of the operators h^ and H^ are shown. Using the spectrum of h^
the essential spectrum of H^ is described. The location of the branches of the essential
spectrum of H^ 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 h^ .
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 H1 and H2 , that is,
H := H1 ® H2 . Consider the linear bounded self-adjoint operators A and B acting on Hi and H2, respectively. Denote by A ® B the tensor product of A and B . Then A ® B is a linear bounded self-adjoint operator on the Hilbert space H . Set T := A ® I1 +12 ® B, where Ii and 12 are the identity operators on Hi and H2, respectively. The operator T is called the tensor sum of the tensor sum of A and
B , and will be denoted by A © B . It can be seen that the operator A©B is well-defined and is in fact linear bounded self-adjoint operator on H . Moreover, for the spectrum of A©B the following equality holds [4]:
cr(T) = cr( A) + cr(B) = {%e R: % = j +A, je cr( A), A e cr(B)}.
(i)
The latter equality implies that, if je^(nsc(A) and A e &disc (B), then
^ + Ae^dsc( A © B).
2
Let us state the problem. We denote by T the two-dimensional torus. The operations
2
addition and multiplication by real numbers elements of T should be regarded as
2 2 operations on R modulo (2xZ Y . For example, if
f = (3^/5,2^/3), g = (4n/5,3^/4) eT2,
then
f + g = (-3^/5,-7^/12) e T2 and 15f = (0,0) e T2.
In the Hilbert space Ls>(T2) of square-integrable symmetric (complex) functions on
2
T , we consider the model operator of the form
Hj := H0 — JV1 — JV2,
where H0 is the multiplication operator by the function s(x) + s(y) :
(Hof Xx, y) = (e( x) + e(y))f (x, y),
and Va , a = 1,2 are interaction operators:
(Vf)(x, y) = J f (x, t)dt, (V2f)(x, y) = J f (t, y)dt
t 1 T 1
Here jj> 0, f e II2((Td)2) ; the dispersion function £(•) is defined on T1 as
s(x) := 1 — cosx.
It is easy to check that under these assumptions the model operator H.. is bounded and
self-adjoint in Is) (T2 ) .
The spectrum, the essential spectrum and the discrete spectrum of a bounded self-adjoint
operator will be denoted by <t(-), ^ess(") and ^disc("), respectively.
We remark that three-particle model operators associated with the system of three particles on a lattice were studied in many works, see e.g., [5-18]. Essential and discrete
spectrum of operator matrices, whose one of the diagonal elements is the three-particle model operators were studied in [19-24]. Following these papers to study the spectral
properties of the operator H , we introduce a bounded self-adjoint operator (so-called Friedrichs model) hj , acting on L2 (T1) by the rule hju := h0 -jv ,
where /q is the multiplication operator by the function £(•) :
(hofXx) = < x)f (x),
and v is the interaction operator on L2T1) :
(vfX x) = I f (t )dt.
T1
From the definitions of H and hu it follows that the operator H can be written as tensor sum
Hj= hj® I +1 ® hj. (2)
Here, I denotes the identity operator on L2 (T1) . Then for the spectrum of H j we obtain that
d(Hu) = o(hj) + o(hj) = {^e R: ^ = a + a,^ e ^/j)}.
We study the spectrum of hj . The perturbation J of the operator /q is a self-
adjoint operator of rank one. 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 hj coincides with the essential spectrum of /q . It is evident that
°(h0) = ^ess(h0) = [0,2]. This yields O-ess(hj) = [0,2].
For any fixed j > 0 we define an analytic function A j (•) (the Fredholm determinant associated with the operator hj ) in C \ [0,2] by
A J Z) := 1 -ji—^—.
^ T1 ) - Z
It follows immediately that the number Z e C \ [0,2] is an eigenvalue of the operator hj if and only if Aj(z) = 0 . Therefore, for the discrete spectrum of hj the equality
^d!sc(hj) = {z e C \ [0,2]: Ajz) = 0} (3)
holds.
From the decomposition
,1214
cos x = 1--x +— x + ...
2! 4!
we
obtain that there exist positive numbers C\, C2 and 8 such that the inequalities
Qx2 < 1 - cosx < C2x2, x e (-8,8)
hold. Therefore from the last estimates using the additive property of the integral we have
dt 8 dt 2 8 dt
J 1-^ J 1-^i-r = .
T11 - cost -8 1 - cost C2 01
The Lebesgue dominated convergence theorem yields A (0) = lim A (z), and
M ri M
z —^—0
hence Am(0) =-ro. Since the function Am(') is continuous and monotonically decreasing on (-^>, 0), the equality
lim Am(z) = 1
z —-X
implies that for any M > 0 the function Am (■) has a unique zero z = Em , lying in
(-OT, 0) . By (3) the number Em is the eigenvalue of hM .
From the positivity of the operator v it follows that the assertion
((Hm - z) f, f) = J (e(t) - z) | f(t) |2 dt - M(vf, f) < 0
t 1
holds for any z > 2, M > 0 and f e L2 (T1) . It means that for all M > 0 the operator
Hm has no eigenvalues lying on the right hand side of 2. Therefore, for all M > 0 the equality
g(Hm) = [0,2] U {Em} (4)
holds with Em< 0.
Taking into account (1), (2) and (4) we can formulate the result about the spectrum of the operator Hm .
Theorem 1. a) For the essential spectrum of Hm the equality
^ss(HM) = [ Em, EM+ 2] U [0,4]
with Em < 0.
b) For any M > 0 the operator H,, has an unique simple eigenvalue, which is equal to 2EM .
Since Em < 0 , it follows that 2E^ < Em .
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ТЕОРЕМА СХОДИМОСТИ ДЛЯ ПОСЛЕДОВАТЕЛЬНОСТИ СИММЕТРИЧНО ЗАВИСИМЫХ СЛУЧАЙНЫХ ВЕЛИЧИН Мамуров Б.Ж.1, Бабакулова С.2
1Мамуров Бобохон Жураевич - кандидат физико-математических наук, доцент; 2Бабакулова Сарвиноз - магистр, кафедра математики, Бухарский государственный университет, г. Бухара, Республика Узбекистан
Аннотация: вероятностные распределения со значениями в пространствах измеримых функций в случае однородных вероятностей определяют условные распределения. А функция распределения со значениями в пространствах измеримых функций есть условная функция распределения. Независимость в этом случае совпадает с условной независимостью. Автором была получена теорема сходимости для последовательности независимых случайных величин, относительно вероятностных мер со значениями в пространствах измеримых функций. В данной работе изучается теорема сходимости для последовательности симметрично зависимых случайных велиин.
Ключевые слова: вероятностное пространство, б - подалгебры, последовательтельность случайных величин, условно независимость относительно б - подалгебры, симметрично зависимость.
Пусть (Т,Е ) - измеримое пространство, т.е. множество Т с выделенной б -алгеброй Еего подмножеств. Через $=$(Т,Е ) будем обозначать совокупность всех измеримых относительно б - алгеброй Е, вещественных функций на пространстве Т. Множество S образует алгебу относительно операций поточеного умножения, сложения функций и поточечного умножения на скаляр. Пусть (Т, Е, т) -вероятностное пространсто, т.е. измеримое пространство (Т, Е) с числовой вероятностной мерой т, определенной на элементах б - алгебры Е. Обозначим через 5™^(Т,Е) идеал алгебры S(T,Е ), которые т- п.в. равны нулю. Фактор-алгебру ¿о (Т,Е, т) = S(T,Е обозначим через Е.
В случаях, когда это не приводит к недоразумению, будем обозначать одной и той же буквой x, функцию х®, teT из S и соответствующий ей класс эквивалентности из Е.
Поточечные алгебраические операции и естественный частичный порядок в S(T,Е) определяют в Е структуру алгебры и линейного упорядоенного пространства над полем вещественных чисел Д1. Нуль и единица алгебры Е обозначаются соответственно через в и 1. Пусть непустое множество и F - некотораяалгеба его подмножеств.
Е-значной вероятностью на измеримом пространстве F ) называется б -аддитивная функция Б : хеЕ : в <х< 1}, такая, что Р(Ц)=1.
В [1] изучены теоретико-вероятностные проблемы для вероятностных мер со значениями в пространствах измеримых функций. Будем придерживается определения и обозначения этой работы. Говорят, что пространство F,P) с Е-значной вероятностью Р обладает свойством полноты: если
McAeF , Р(А)= в то MeF и Р(М)= в .
