eu ON THE FORM OF CORRELATION FUNCTION FOR A CLASS OF NONSTATIONARY FIELD WITH A DISCRETE AND MIXED SPECTRUM

Abstract: The present paper is devoted to the derivation of an explicit form of linearly representable random fields in the form h(x1, x2) = exp {i(x1A1 + x2A2)} h, where h ∈ H, H is a Hilbert space, operators A1, A2 are such that A1A2 = A2A1 and C 3 = 0 where C = A∗1A2 −A2A1. The results obtained are the generalization of theorem proved in [3], [5], [7]. It is shown that a rank of nonstationary of field h(x1, x2) depends not only on a degree of non-self adjoint of A1, A2 but on a degree of nilpotency of commutator C(C3 = 0). In the present paper an explicit form of correlation function for discrete spectrum of A1 and A2 is derived. A form in the case of spectrum of operator A1 is constructed in zero and that of the operator A2 is pure discrete of A1and A2 is zero and the other is discrete, is obtained.

Consider a vector field h(x) where x = (x 1 , x 2 ) ∈ IR 2 , with values in Hilbert space H.In this paper we suppose that h(x) = Z x h where Z x = exp [i(x 1 A 1 + x 2 A 2 )].In this case A 1 and A 2 are such operators in the Hilbert space H for which A 1 A 2 = A 2 A 1 , and the operator function Z x is called a two-parametric commutative semigroup.The main tool of correlation theory for vector fields in a Hilbert space H is a correlation function [3]: where x, y ∈ IR 2 , for twice permutational classes of linear operators {A 1 , A 2 }, . Generalizing the results given [3,5,7] has introduced partial infinitesimal correlation functions (ICF) by relations (under the assumption that K(x 1 , x 2 , y 1 , y 2 ) is twice differentiable function): where W 1 , W 2 and W are not independent.Indeed: Similarly it is easy to get Recall that the field h(x) in H is called dissipative if (A 1 ) I ≥ 0. As in onedimension case it is easy to establish [3,7] that lim If the correlation function depends only on the difference of arguments then a field is called a stationary field [3] (just in this way a stationary was defined by Kolmogorov, A.N.).Then (3) may be presented in the form , where K ∞ (x − y) is a Hermitian-positive function which may be considered as a stationary field correlation function, K 1 ∞ (x 1 − y 1 , x 2 , y 2 ) (as well as K 2 ∞ (x 1 − y 1 , x 2 , y 2 )) with variable x 1 − y 1 is a Hermitian-positive function for each x 2 , y 2 , and , as a function of x 2 , y 2 , it is a dissipative curve of one variable in H. Thus, it is determined by the infinitesimal correlation function W (x, y).

1.2.
Recall [1,2] that the rank of nonstationary function h(x) of twice permutational system of linear operators A 1 , A 2 is the greatest rank of all the quadratic form n α,β=1 It is not difficult to show that the rank of nonstationarity for the present case coincides with the dimension of the space as in [3]) and in addition Derivation of formula (6): From formula (2) it follows that Therefore, then we get that Since A 1 and A 2 are twice permutable then, In case dim H 0 = 1, i.e. when the rank of nonstationarity of vector field h(x) is equal to one we get where Φ(x) = h(x), h 0 .

Correlation Functions and Spectral Representation of the Twice
Permutational Fields of Rank 1 2.1.

Consider the vector field
As it was shown in [5,7], the ICF of vector field h(x 1 , x 2 ) has the form where Γ is a closed path that contains all the spectrum of operator A , one can represent the function Φ(x 1 , x 2 ) in the form The closed path Γ k includes the spectrum of the operator A k (k = 1, 2).While calculating integrals in (8) one can pass to any system of operators acting in Hilbert space • H, which are unitary equivalent to the original operators

2.2.
Consider the case, when the field h(x 1 , x 2 ) belongs to the class K (1) 22 [7], i.e. the spectrum of each of the operator A k (k = 1, 2) is pure discrete.In this case, the model space where is a sequence of non-real points of spec- The operators • A 2 can be represented as follows Since the intersection of non-Hermitian subspaces of operators • A 2 coincides with the subspace of functions that does not depend on the arguments then • h 0 (p, q) ≡ 1, and Then there exists a representation p − λ 1 )(λ Thus, where, the functions Λ p (x 1 ) and Λ (2) q (x 2 ) are defined as follows p − λ 1 ) Similar to, the class of twice permutable system of linear operator of the vector field

Correlation Functions of Commutative
We introduce the correlation functions It is not difficult to see that for the case of the vector field h(x) one can obtain [4] Here the operator D is self-adjoint and is of the form Let us show that D may be represented as (12).From formula (10) using differentiation rules we can easily get Then we can find This completes the proof (12).Elementary evaluations show that the operator D in (12) can be reduced to In order to obtain a concrete form of operator D we confine ourselves, from now on, to systems of linear operators that satisfy the following theorem which is proved in [4].A system of operators A 1 , A 2 is called a simple system [3] if there is no subspace in H, which reduces the operators A 1 and A 2 , a contraction on which is self-adjoint at least for one of the operators A k .
Theorem 3.1.[4] Let us assume that a simple commuting system of linear operators A 1 , A 2 is such that: Then, the space H is decomposed into orthogonal sum In what follows, we assume that a system of linear operators {A 1 , A 2 } satisfies the assumptions of Theorem 3.1, to conclude the result stated in lemma 3.1, below.Let Obviously, this follows from the condition C 3 = C * 3 = 0.One can easily see that Let us denote by h 1 as a vector such that {λ h1 } = H 1 H 0 and introduce the following vectors: where the vector g 1 is such that g 1 + h 3 + g 2 + g 3 = h 0 , and h 0 is a basis vector of space H 0 .
Then it is easy to see that Thus, the operator D, corresponding to defect of non-stationary, maps H into a six-dimensional space.
Let us find an explicit form of self-adjoint operator D defined in H D .Really, it is easy to see that where (A 1 ) I h 3 = α 3 h 3 .Therefore, Dh 3 , g 2 = 0 and Dh 3 , g 3 = 0.
By repeating the same arguments one can obtain Hence, we have proved the following lemma. Lemma where d α,β ∈ IR are real numbers.

3.2.
Let us consider the infinitesimal correlation function W (x, y)(11): Then, by virtue of (15) one can obtain Denote by Let us find the form of functions Φ α (x).First, note that the functions Φ α (x) are invariant under unitary equivalence and hence we can use the presentation model which is derived in [7].As is obvious from the models the vector-functions exp [−i( ]ℓ α generate subspaces L α invariant under the operators A * 1 and A * 2 where the restrictions of the operators A * 1 and A * 2 to L α are twice permutable.Denote the images of the vectors {ℓ α } under unitary equivalence (which is realized by the model restriction) by {h α }, and the image of h by f (p, q), where the functions f (p, q) are defined in the following domain Then, using the results of Section 2, we can get an explicit expression for the functions Φ α (x 1 , x 2 ) , where 1 < N 1 < N 2 < N 3 and 1 < M 1 < M 2 < M 3 and the functions Λ p (x 1 ), Λ q (x 2 ) are given by Λ (1)  p (x 1 ) = − p − λ 1 ) q − λ 2 ) and at last β q .Thus, we can formulate the following theorem.Theorem 3.2.Assume that a system of linear operators {A 1 , A 2 } satisfies to the assumptions of theorem 1 where the spectrum of each operator A k (k = 1, 2).Then the infinitesimal correlation function W (x, y) (11) is represented in the form (16) where d α,β ∈ IR and the functions Φ α (x) are defined in (17).
To evaluate d α,β we represent ℓ α graphically into the figures where ℓ α 's are constants in the indicated areas.
So that where χ Dα is the characteristic function of the domain D α , which is shown in the pictures for ℓ α and S Dα is the area.For example, Let us define Dℓ 1 : After (A 2 ) I we get Therefore, .
The other coefficients are similar.Let the field h(x 1 , x 2 ) belongs to the class K (1) 12 [7], i.e. the spectrum of operator A 1 is concentrated in zero and that of operator A 2 is pure discrete.A model space • H for this case is a totality: It is easy to see that in this case where functions Λ k is defined in Sections 3.2 and is the Bessel function of zero order.

4.2.
One may deduce the results analogous to that of Section 3, for Theorem 3.1 and Lemma 3.1 where W (x, y) is in the form, given in (16).The functions f k (y) defined in the following domain q − λ 2 ) q and is the Bessel function of zero order.
Thus, we can formulate the following theorem.
Systems of Operators in Case of the Nilpotentness the Commutator C = [A * 1 , A 2 ](C 3 = 0) with a Discrete Spectrum 3.1.

4 .
Correlation Functions of Commutative Systems of Operators in Case of the Nilpotentness the Commutator C = [A * 1 , A 2 ](C 3 = 0) with a Mixed Spectrum 4.1.
3.1.The matrix of the operator D in the basis {h 1 , h 2 , h 3 , g 1 , g 2 , g 3 } of the space H D can be written by the form 11 d 12 d 13 d 14 0 0 d 12 d 22 d 23 d 24 d 25 0 d 13 d 23 d 33 d 34 d 35 d 36 d 14 d 24 d 34 d 44 d 45 d 64 0 d 25 d 35 d 45 d 55 d 56 0 0 d 36 d 46 d 56 d 66