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THRESHOLD EIGENVALUES OF A TWO-CHANNEL
MOLECULAR-RESONANCE MODEL Nematova Sh.B.1, Rasulov T.H.2 (Republic of Uzbekistan) Email: Nem [email protected]
'Nematova Shohida Bobojon qizi — Student; 2Rasulov Tulkin Husenovich — PhD in Mathematics, Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: in the present paper we consider a two-channel molecular-resonance model. This model can be represented as a bounded and self-adjoint 2 X 2 operator matrix A acting in the two-particle cut subspace of standard Fock space. It is associated with the Hamiltonian of a system consisting at
most two particles, interacting via both a pair contact potentials ^ 0) and creation and
annihilation operators (^ > 0) . We construct the Fredholm determinant corresponding to A . We find some necessary and sufficient conditions for that the number z = min CTess (A) to be an eigenvalue (threshold eigenvalue) of A .
Keywords: threshold eigenvalue, molecular-resonance model, coupling constant, nuclear space, molecular space, Hamiltonian, operator matrix, essential spectrum.
ПОРОГОВЫЕ СОБСТВЕННЫЕ ЗНАЧЕНИЯ ДВУХКАНАЛЬНОЙ МОЛЕКУЛЯРНО-РЕЗОНАНСНОЙ МОДЕЛИ Неъматова Ш.Б.1, Расулов Т.Х.2 (Республика Узбекистан)
'Неъматова Шохида Бобожон кизи — студент; 2Расулов Тулкин Хусенович — кандидат физико-математических наук, заведующий кафедрой,
кафедра математики, Бухарский государственный университет, г. Бухара, Республика Узбекистан
Аннотация: в настоящей работе мы рассмотрим двухканальную молекулярно-резонансную модель. Этамодель представляется как ограниченная и самосопряженная 2 X 2 операторная матрица A действующая в двухчастичном обрезанном подпространстве стандартного фоковского пространства. Оно ассоциировано гамильтониановской системой, состоящей из не более чем двух частиц, взаимодействующих как с помощью парного контактного
потенциала ^ 0), так и с помощью операторов рождения и уничтожения (^ > 0) .
Построим определитель Фредгольма соответствующий к A. Найдем необходимые и достаточные условия для того, чтобы число z = min CTess (A) являлось собственным
значением (пороговые собственное значение) оператора A .
Ключевые слова: пороговые собственное значение, молекулярно-резонансной модель, параметр взаимодействия, ядерное пространство, молекулярное пространство, Гамильтониан, операторная матрица, существенный спектр.
We adopt the following conventions the present paper. Let C be the field of complex numbers, T3 be the three-dimensional torus, the cube (- 7, 7] with appropriately identified sides
equipped with its Haar measure and L2 (Г ) be the Hilbert space of square integrable (complex) functions defined on 1 . We consider a two-channel Hilbert space H := H0 © Hl consisting of a one-dimensional "molecular" space H0 := C (channel 1) and a "nuclear" space (channel 2). The
elements of H are represented as vectors f = ( f0,fj), where f G HQ and f G Hj. The inner product
f,g)H := f0g0 +(^gi)
of any two elements f = (f0, fj), g = (g0, gj )g H, is naturally defined via the inner products f0 g0 in H0 and ^fj, g^ in Hthat is
(fl,H :=i¿(O&tfdH
T 3
As a Hamiltonian in the Hilbert space H we consider the 2 X 2 block operator matrix
A:=
r A A ^
V A01 A11 y
(1)
where the matrix elements A : H ^ H , i < J, i, J = 0,1 are defined by
A00 f0 ^/p A01 ./1 = f J C^ > f\C^ , ,
T3
( A11/1 )( p )= C (P )/1 ( P ) - f 2 V2 CP)f 1 J V2 (t )/1 (t .
T 3
Here fla, = 1,2 are real numbers, CO0 G ^ is a trial "molecular" energy, a vector V1 G H1 provides the coupling between the channels and An is the (self-adjoint) "nuclear Hamiltonian" in H . We assume that the functions Ua C'),^ = 1,2 and C1 (') are real-valued continuous functions on T 3 .
Under these assumptions the operator A is bounded and self-adjoint.
It should be mentioned that the Hamiltonian (1) is a rather more general model than that one considered by Friedrichs [1] (for a some discussion of An see, e.g., [2,3] and the references cited therein).
Molecules are usually treated as purely Coulombic systems, while the strong interaction between their nuclear constituents is assumed to play a negligible role. If there is no coupling between the
channels for D1 = 0 the spectrum of A consists of the spectrum of Au and the addition
discrete eigenvalue Co . A nontrivial coupling between the channels will, in general, shift the
eigenvalue Co into an unphysical sheet of the energy plane. The resulting perturbed energy appears
as a resonance, i.e., as a pole of the analytic continuation of the resolvent ( A - z) 1 taken between suitable states (see [4]). In [5], the Hamiltonian (1) was studied in order to exhibit explicitly the mechanism leading to the enhancement of fusion probability in case of a narrow near-threshold nuclear resonance.
Let the operator A0 acts in H as
A :
r 0 0 ^
with Ulf )(p)= (p(P)
0 • 0 A0
{ 0 A11J
The perturbation A - A of the operator A0 is a self-adjoint operator of rank 3, and thus, according to the Weyl theorem, the essential spectrum of the operator A coincides with the essential
spectrum of A0 . It is evident that C7ess (A0 ) = [W; M], where the numbers W and MM are defined by
m:=min (ol (p ), M:=max®1( p ).
x ^ T x ^ T
This yields (Tess (A) = [w; M].
For any fixed Li , X = 1,2 we define an analytic function A (•) (the Fredholm
* x ^ ^ Li L2
determinant associated with the operator A ) in C \ [w; MM] by
A^ (z) := A^ (z) (z) + ( A(3) (z))2,
where the functions A(;) (•), A(2) (•) and A(3) (•) are defined by
A™(z):=c -zJuU^,
1 ) - z
A(2) (z):=1-^2 r-^-, A(3)(z):= J u (t)u(td.
A ^c^t) - z Тз c(t) - z
By the Birman-Schwinger principle and the Fredholm theorem we conclude that the operator A has an eigenvalue z e C \ Tm; M] if and only if A ( z) = 0.
Suppose that the function ¿¿^ (•) has an unique non-degenerate minimum at the point po e T3, the functions ^ (•) , V^= 1,2 have continuous partial derivatives up to order 3 inclusive at some neighborhood of the point po eT3 and Lebesques measure of supp{Ll(-)}o supp{L>2(-)} is equal to zero, where supply (•)} of the function L>a (•) .
u;(t)dt
For the case C0 > m we set
(co - m)
, u;2 (t )dt л
j co; (t) - m
2/
Ao :=
J
c 1 (t) - m
Main result of the paper is the following theorem
Theorem 1. The number Z = m is an eigenvalue of A with multiplicity 2 if and only if
Ma = M°a and va (p0) = 0 for a = l,2.
This result plays important role in the investigations of the essential and discrete spectrum of the corresponding 2 X 2 (see e.g., [6-13]), 3 X 3 (see e.g., [14-23]) and 4 X 4 (see e.g., [24, 25]) operator matrices in the cut subspaces of standart Fock space.
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