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INVESTIGATION OF THE SPECTRUM OF A DIAGONALIZABLE 4x4-OPERATOR MATRIX Mustafoeva Z.E.1, Rasulov T.H.2 (Republic of Uzbekistan) Email: [email protected]
'Mustafoeva Zarinabonu Erkin qizi — Master Student; 2Rasulov Tulkin Husenovich — PhD in Mathematics, Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: in this paper we consider a 4 X 4 operator matrix A acting in the direct sum of four Hilbert spaces is considered. It is unitarily equivalent to 2 X 2 diagonal operator matrix diag{A(+), A(-]} . Diagonal elements A , s = + , are acting in the four-particle cut subspace of Fock space. Essential and point spectrum of A are described using spectrum of more simple two 4 X 4 operator matrices A(s\ s = + . We describe the location of the branches of essential spectrum of A. It is shown that the essential spectrum of A consists the union of at most 7 bounded closed intervals. An inclusion for the discrete spectrum A is obtained.
Keywords: operator matrix, standard Fock space, generalized Friedrichs model, essential spectrum.
ИССЛЕДОВАНИЕ СПЕКТРА ОДНОЙ ДИАГОНАЛИЗИРУЕМОЙ 4Х4-ОПЕРАТОРНОЙ МАТРИЦЫ Мустафоева З.Э.1, Расулов Т.Х.2 (Республика Узбекистан)
'Мустафоева Заринабону Эркин кизи — магисрант; 2Расулов Тулкин Хусенович — кандидат физико-математических наук, заведующий кафедрой,
кафедра математики, Бухарский государственный университет, г. Бухара, Республика Узбекистан
Аннотация: в данной работе рассматривается 4 X 4 -операторная матрица А, действующая в прямой сумме четырех гильбертовых пространств. Оно унитарно
эквивалентно к 2 X 2 диагональному операторному матрицу diag{A^; A( .
Диагональные элементы A(s), s = +, действуют в четырех-частичном урезанном подпространстве фоковского пространства. Существенный и точечный спектр оператора А описываются с помощью спектров двух более простых 4 X 4 -операторных матриц A(ss = + . Мы описываем местоположение ветвей существенного спектра оператора A . Установлено, что существенный спектр оператора A^s), s = + состоит из объединения не
более чем семь отрезков. Получено включение для дискретного спектра оператора А . Ключевые слова: операторная матрица, стандартное пространство Фока, обобщенная модель Фридрихса, существенный спектр.
In statistical physics [1], solid-state physics [2] and the theory of quantum fields [3], one considers systems, where the number of quasi-particles is bounded, but not fixed. Often, the number of particles can be arbitrary large as in cases involving photons, in other cases, such as scattering of spin waves on defects, scattering massive particles and chemical reactions, there are only participants at any given time, though their number can be change.
Recall that the study of systems describing N 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 H(N) of Fock space, consisting of n < N particles. In [4] geometric and commutator techniques have been developed in order to find the location of the spectrum and to prove absence of singular continuous spectrum for Hamiltonians without conservation of the particle number.
In the present paper we consider a 4 X 4 operator matrix A . It is unitarily equivalent to the 2 X 2 diagonal operator matrix diag } . The diagonal elements A s , s = ± are acting
in the four-particle cut subspace of standard Fock space. We investigate the essential, point and discrete spectrum of A .
Let C be the field of complex numbers, Td be the d-dimensional torus, L2 ((Td )n ) be the
Hilbert space of square integrable (complex) functions defined on (Td) , n g N and F(L2 (Td )) be the standard Fock space over L2 (Td ) , where
F(L2(Td )):= C © L2(Td ) © L2((Td)2) ©....
Set 3 := C2 ® F(L2(Td)) . We write elements F of the space 3 in the form
F = {f(s), fS)0U f2.S)(K, k2),..., f s)(k„ k2,..., kn ),...}
of functions of an increasing number of variables (k1,..., kn ) G (T d ) n, and a discrete variable S = ± . The norm in 3 is given by
IF2 :=I|fo(sT + ZJ (Td f^,.., kn )\2 dkv..dkn .
s=± s,n
Let F {m)(L2(Td )):= C © L2(Td ) © L2(Td )2) ©... © L2 ((Td )m ), m g N .
We consider the operator A acting the Hilbert space
33 := C2 ® F(3)(L2(Td)) and
represented as a tridiagonal 4 X 4 operator matrix
A :=
with elements
<4oc/oW = ¿So", Aoi/i^ = aj Tdv(q l)ff\q ')dq (An/1(s)){ p) = (ss + w( p))/ s)( p), (Ai2/2(s) )(p) = aj Td v(q' )/ts) (P, q' )dq' ,
(A/)(p, q) = (ss + w( p) + w(q))/^s'> (p, q), ()(p) = aj Td v(q' )/ts) (p, q, q')dq'
(A3/ s))(p, q, t) = (ss + w( p) + w(q) + w(t ))/« (p, q, t),
{/o( s), /i( s), /2(s), /(s), s = +}e33
Here Aj is the adjoint operator to Aj for i < i and the norm of element
/ = {fo(s),fi(s),/2(s),/3(s),s = +}e33 is defined by
2 2 l|/||2 =X (|/o(sT +j / p)|2 dp +j (Td f\/(\p, q)\ dpdq + j (Td/\p, q,t)\ dpdqdt) S
s=±
is a real number, v(-) and w(-) are the real-valued continuous functions on Td , and a > 0 is the coupling constant.
To investigate the spectral properties of A we introduce the following two bounded self-adjoint operators As),s =+ , which acts in FL2(Td)) as
A Aoo Ao1 0 0 1
A A01 A11 A12 0
0 A A12 A22 A23
,0 0 A A23 A33 y
f A s ) aoo a01 0 0 ]
A* a01 A( s ) a11 a12 0
0 A* A( s ) a23
V 0 0 A* a23 A( s) ^33 y
A :=
with the entries (« f -
Ao(0/ = sefo, Aifi = «j v(q1 )/ (q' )dq',
(A/^Xp) = (^ + w(p))/ (p), (Ai2/i2)(p) = a\ Td v(q1 )/ (p, q1 )dq1
^ (A2)f2 )(p, q) = (~se + w( p) + w(q))f2 (p, q),
(A23f3)(p, q)=«j Td v(q' )f3(p, q, q /)dq1,
(A(3s)f3)(p, q, t) = (-se + w( p) + w(q) + w(t ))f3 (p, q, t), (fo,fi,f2,f() e F(3)(L2(Td)).
Using simple calculations we obtain the following equalities
(iifo)(p) = «v( p)fo, (A^/iX p, q) = «v(q)fi(p), (A^Xp,q,t) = «v(t(p,q)^Cfo,fi,f2) e Fi2)(L2(Td))
A,, Ai and A,, resp A
A12 and A23 are called
We remark that the operators aoi, a12 and
/-L9 3 ; /J-oi
annihilation resp. creation operators, respectively. In this paper we consider the case, where the number of annihilations and creations of the particles of the considering system is equal to i. It means
that Aj = o for all |i — j > i. The spectral properties of 2x2, 3x3 and 4x4 operator matrices are
studied in many works, see for example, [5-23]. Spectral inclusion property for diagonally dominant unbounded operator matrices is investigated in [24].
Denote by <x(-),<ess ("X^ppG) and <Xdisc('), respectively, the spectrum, the essential
spectrum, the point and the discrete spectrum of a bounded self-adjoint operator.
The following theorem describes the connection between spectra of A and A(s), s = ± .
Theorem 1. The equality <(A) = <(A(+)) u<(A(—*) holds. Moreover,
<ess (A) = <ss (A(+) ) U <ess (A- ) , <p (A) = <p(A«) U < (A« ) .
The set <Jess (A(s)), S = ± consists at most seven bounded closed.
Since the part of ^disc(A^s) ) can be located in <Xe
( A) Œ^diSc( A(+)) A")
we have the inclusion
disc^-
^disc(A) = {^disc(A(+) ) ^ ^disc(A(-))} \ ^ess
More exactly
^disc( A) = U{^disc( A(s))\^ess( A(-s))}.
(1)
( A)
(2)
We remark that for S = + the operator A(s) has more simple structure than A , and hence, Theorem 1 and relations (1), (2) plays important role in the next investigations of the spectrum of A.
s
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