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PHYSICO-MATHEMATICAL SCIENCES
ESSENTIAL SPECTRUM OF A 2x2 OPERATOR MATRIX AND THE
FADDEEV EQUATION Dilmurodov E.B.1, Rasulov T.H.2 (Republic of Uzbekistan) Email: [email protected]
'Dilmurodov Elyor Baxtiyorovich — PhD Student; 2Rasulov Tulkin Husenovich — PhD in Mathematics, Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: we consider a 2 X 2 operator matrix аи, и>0 acting in the direct sum of one- and two-particle subspaces of a bosonic Fock space. It is related with the system of non conserved number of quasi-particles. We obtain an analogue of the Faddeev equation for the eigenfunctions of . We
describe the location of the essential spectrum of An via the spectrum of a family of generalized
И
Friedrichs models. It is shown that the essential spectrum of An consists the union of at most 3
И
bounded closed intervals. We introduce new branches of the essential spectrum of Аи .
Keywords: operator matrix, bosonic Fock space, generalized Friedrichs model, essential spectrum, the Faddeev equation.
СУЩЕСТВЕННЫЙ СПЕКТР ОДНОЙ 2Х2 ОПЕРАТОРНОЙ МАТРИЦЫ
И УРАВНЕНИЕ ФАДДЕЕВА Дилмуродов Э.Б.1, Расулов Т.Х.2 (Республика Узбекистан)
'Дилмуродов Элёр Бахтиёрович — базовый докторант; 2Расулов Тулкин Хусенович — кандидат физико-математических наук, заведующий кафедрой,
кафедра математики, Бухарский государственный университет, г. Бухара, Республика Узбекистан
Аннотация: рассматривается 2х 2 операторная матрица Ац, и> 0, действующая в
прямой сумме одно- и двухчастичного подпространства бозонного пространства Фока. Оно связано с системой несохраняющихся чисел квазичастиц. Получен аналог уравнения Фаддеева
для собственных функций оператора Аи. Местоположение существенного спектра
И
оператора Ац описано с помощью спектра семейства обобщенных моделей Фридрихса. И
Показано, что существенный спектр оператора Аи состоит из объединения не более трех
И
отрезков. Вводим новые ветви существенного спектра оператора Ац .
И
Ключевые слова: операторная матрица, бозонное пространство Фока, обобщенная модель Фридрихса, существенный спектр, уравнение Фаддеева.
It is well-known that every bounded linear operator acting in the direct sum of two Hilbert spaces always admits 2х 2 block operator matrix representation [1]. The problems related with such matrices arise in statistical physics [2], solid-state physics [3] and the theory of quantum fields [4].
In the present paper we consider the family of 2 X 2 operator matrices Аи ( И > 0 is a
coupling constant) associated with the lattice systems describing two identical bosons and one particle,
another nature in interactions, without conservation of the number of particles. This operator acts in the direct sum of zero-, one- and two-particle subspaces of the bosonic Fock space and it is a lattice analogue of the spin-boson Hamiltonian. We derive an analogue of the Faddeev type system of
integral equations for eigenvectors of An . We describe the location of the essential spectrum
№
C'ess (A^ ) of A№ , via the spectrum of a family of generalized Friedrichs models A^ (k) , k e Td . We introduce a new branches of C ( A, ) and show that it consists the union of at
most 3 bounded closed intervals. We find the discrete spectrum of An .
№
Let Td be the d -dimensional torus, the cube (_K,K]d with appropriately identified sides equipped with its Haar measure. Let ^2(Td) be the Hilbert space of square integrable (complex) functions defined on Td and L¡2((Td)2) be the Hilbert space of square integrable (complex) symmetric functions defined on (Td )2. Denote by H the direct sum of spaces
Hx := L2(Td) and H2 := Ls2((Td)2), that is, H := H1 ©H2. The
spaces H and
H2 are called one- and two-particle subspaces of a bosonic Fock space Fs (L2(Td)) over L2(Td ), respectively.
Let us consider a 2x 2 operator matrices A№ acting in the Hilbert space
H as
_ 4)0 M)1
A№
M)i Aii y
with the entries
(Af)(P) = w1(k)f1(p), (AJ2)(p) = Jv(s)f2(p,s)ds,
T"
(Aifi)^q) = q)/¡Cp.q\ fE H,, i = 1,2,
where fl > 0 is a coupling constant, the functions W1 (•) and W2 (•; •) are real-valued continuous functions on Td and (T ) respectively. In addition the function W2(^) is a symmetric, that is, W2(p; q) = W2(q; p) for any p, q E T" . Under these assumptions the operator Aj is bounded and self-adjoint.
Let Ho := C . To study the spectrum of the operator Aj we introduce a family of bounded self-adjoint operators (generalized Friedrichs models) Aj(k), k E Td which acts in Ho © H1 operator matrices
A00 (k ) M)1 ^
v M0l All(k)y with matrix elements
Ao(k)fo = Wi(k)fo, (A0f1) = Jv(t)f()dt, (Aiif2)(P) = W2(kP)fl(P).
Td
According to the Weyl theorem, for the essential spectrum of the operator A, (k), we have C7ess(A,(k)) = [m(k);M(k)],where the numbers m(k) and M(k) are defined by
m(k) := min W2(k,p) and M(k) := max W2(k,p).
For any fixed k £Td we define an analytic function A^ (k; •) (the Fredholm determinant associated with the operator A, (k) ) in C \ [m(k);M(k)]
a n a n^ V2 t V2(t)dt A,(k; z) := Wi(k) - z w
2 fdW2(k,t) - z
Set m := mind W2 (p, q), M := max W2 (P, q) and A, := U ad*c((k)).
p,q£T pq£T kcTd
k£Td
We introduce a operator T^ (z) acting in H1 as
(T„(z)g)(p) = -f^ J ^^• z= [m;M]uA,.
M 2A,( p, z) TdW-( p, t) - z
The following theorem [5-15] is an analog of the well-known Faddeev's result for the operator An . Theorem 1. The number z £ C \ S is an eigenvalue of the operator An if and only if the
/1
number X = 1 is an eigenvalue of the operator T'u (z). Moreover the eigenvalues z and 1 have the same multiplicities.
We point out that the equation T^ (z)p = p is an analogue of the Faddeev type integral equation for eigenfunctions of the operator An .
Now we describe [11, 12, 16, 17] the location of the essential spectrum of the operator An by the spectrum of the family A, (k) of generalized Friedrichs models.
Theorem 2. For the essential spectrum of A, the equality CTess (A,) = Sn holds.
Moreover the set A consists no more than three bounded closed intervals.
M
Following we introduce the new subsets of the essential spectrum of An.
h1
Definition 1. The sets A,, and
[m; M ]
are called two- and three-particle branches of the
essential spectrum of A, , respectively.
The definition of the set Au and the equality U[m(k);M(k)] = [m;M] together with
k£Td
Theorem 1 give the equality
(A,) = Ua( A,(k)). (1)
k£Td
Here the family of operators A, (k) have a simpler structure than the operator A,. Hence, in many instance, (1) provides an effective tool for the description of the essential spectrum. The
spectral properties related with the threshold analysis of a family of 2 X 2 operator matrices were studied in [18-23]. In the paper [24] spectral inclusion property for diagonally dominant nxn unbounded operator matrices was studied.
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