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STRUCTURE OF THE NUMERICAL RANGE OF FRIEDRICHS MODEL WITH RANK TWO PERTURBATION Bahronov B.I.1, Rasulov T.H.2 (Republic of Uzbekistan) Em ail: Bahronov451@scientifictext.ru
'Bahronov Bekzod Islom o 'g 'li - 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 Friedrichs model H with rank two perturbations. It is associated with a system of two quantum particles on d— dimensional lattice. We investigate the structure of the closure of numerical range W(H) of this operator in detail by terms of its
parameters for all dimensions d of the torus T . For d > 3 and X — 1,2 we find the critical value Ha of the parameter Ja > 0 such that for all J( £ (0;j(] there is no eigenvalues
outside of the essential spectrum of H and for all fj,a > Ja there are two eigenvalues of H .
Moreover, we find the conditions, which guarantees that the set W (H) is closed.
Keywords: Friedrichs model, perturbation, quantum particles, numerical range, spectrum, threshold
eigenvalues, virtual level.
СТРУКТУРА ЧИСЛОВОЙ ОБЛАСТИ ЗНАЧЕНИЙ МОДЕЛИ ФРИДРИХСА С ДВУМЕРНЫМ ВОЗМУЩЕНИЕМ Бахронов Б.И.1, Расулов Т.Х.2 (Республика Узбекистан)
'Бахронов Бекзод Ислом угли — магистрант; 2Расулов Тулкин Хусенович - кандидат физико-математических наук, заведующий кафедрой,
кафедра математики, Бухарский государственный университет, г. Бухара, Республика Узбекистан
Аннотация: в работе рассматривается модель Фридрихса H с двумерным возмущением. Оно ассоциировано с системой двух квантовых частиц на d —мерной решетке. Структура замыкания числовой области значений W(H) этого оператора подробно исследована в
терминах его параметров при всех размерностях d тора Td . Для d > 3 и a —1,2
находим критическое значение Ца параметра JUa > 0 такое, что при всех
Ja £ (0; Jx ] не существуют собственные значения, лежащие вне существенного
спектра оператора H и для любых Ja > Ja оператор H имеет два собственных значения. Найдем условие, гарантирующее замкнутость множества W(H) . Ключевые слова: модель Фридрихса, возмущения, квантовые частицы, поля значений, спектр, пороговые собственные значение, виртуальный уровень.
In a well-known monograph [1], Friedrichs considered the operator of multiplication by an independent variable as an unperturbed operator and chose a perturbation to be given by an integral operator. This model was subsequently called the model of the perturbation theory of the continuous spectrum because the continuous (essential) spectrum of a self-adjoint operator is unchanged under compact perturbation. The present paper is devoted to the study the numerical range of a Friedrichs model, where as a perturbation chosen an integral operator with rank two.
First, we recall the definition of numerical range. Let L be a complex Hilbert space with inner product (•; •) and H be a linear operator in L with domain D(H) C L. Then the numerical
range of an L is the subset of the complex numbers C, given by
W(H) := {(Hx, x) : x £ D(H), ||x|| = 1} . It was first studied by O.Toeplitz in [2]. In [3]
F.Hausdorff showed that indeed the set W(H) is convex. We recall that the numerical range of a bounded linear operator satisfies the so-called spectral inclusion property
W(H) c up (H), W(H) c u(H)
for the point spectrum U (H) (or set of eigenvalues) and the spectrum u(H) of H ; note
that W(H) is closed if dim L <<n .
The notion of numerical range is generalized by the different ways, see for example [4-7]. One important use of W(H) is to bound the spectrum u(H) .
For positive integer number d let Td be the d -dimensional torus, the cube (—ff, ff] with
appropriately identified sides equipped with its Haar measure and L2(Td ) be the Hilbert space of
square integrable (complex) functions defined on Td .
Let us consider a so-called Friedrichs model H acting on the Hilbert space L2 (Td )
H := H0 — juV + u2V2, (1)
where the operators H0 and V , X = 1,2 are defined by
(Hof)(p) = u(p)f (p), (Vaf)(p) = uava(p)\Td va(t)f (t)dt, x = 1,2.
Here ja > 0, X = 1,2 are positive reals, u(-) and Vi (•) , i = 1,2 are real-valued
continuous functions on Td . Under these assumptions, operator H defined by (1) is bounded and self-adjoint.
Notice that when we study the model operators associated to a system of three particles on ad -dimensional lattice [8-19] and interacting via non-local potentials, the role of a twoparticle discrete Schrodinger operator is played the Friedrichs model. Such type operators are also important in the investigations of the essential spectrum and the number of eigenvalues, located inside (in the gap, in the below of the bottom) of the essential spectrum of operator matrices [20-25].
It is clear that the essential spectrum of the operator H coincides with the spectrum of [m;M] , where the numbers m and M are defined by m := min u(p), M := max u(p).
p£Td p£Td
Throughout this paper, we assume that the function u(-) has the continuous partial derivatives up
to the third-order inclusive on Td , in addition it has an unique non-degenerate minimum at the point px £ T3 and an unique non-degenerate maximum at the point p2 £ T3. For the case d > 3 it is easy to check that the integrals
, vl(t)dt , vl(t)dt *Tdu(t) — m' >TdM — u(t),
are positive and finite. In this case we introduce the following quantities: j- uf(t )dt ^Td u(t) — m
0 _ f f ^12(t)dt \' 0 _ ^ u22(t)dt vl
j1:=|JT"u(t)- m ' U2:=|Jl
jTdM — u(t)
One can show that under the condition
mes(supp {u1(-)} o supp {u2(-)}) = 0. (3)
for any /a < /1°, C = 1,2 the operator H has no eigenvalues lying in the outside of [m;M] and for any / > /1°, C = 1,2 the operator H has two simple eigenvalues e1 e (-ro; m) and E2 e (M; ro).
The following theorem describes the numerical range W(H) of H . Theorem 1. Let d > 3 and the condition (3) be fulfilled.
1) If J < J0 for any a — 1,2, then w(h) — [m;m] — <r(H);
2) If j < j0 and j2 > j20, then w(h) — [m; E2] с );
3) If j > j0 and j2 < J0, then W(H) — [E:;m] с a(h);
4) If j > J0 for any a — 1,2, then W(H) — [ex;E2] с ).
A simple calculations show that if d — 1,2 and Ua (pa ) ^ 0, a — 1,2, then the integrals (2) are infinite. Therefore, for any Ja > 0, a — 1,2 the operator H has two simple eigenvalues Ej £ (-да; m) and e2 £ (m; да). From these fact it follows that
w (h) — E E2] c^(H).
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