SPECTRAL ANALYSIS FOR A CLASS OF INTEGRAL-DIFFERENCE OPERATORS: KNOWN FACTS, NEW RESULTS, AND OPEN PROBLEMS

We present state of the art, the new results, and discuss open 
problems in the field of spectral analysis for a class of 
integral-difference operators appearing in some nonequilibrium 
statistical physics models as collision operators. The author 
dedicates this work to the memory of Professor Ilya Prigogine, 
who initiated this activity in 1997 and whose 
interesting and most enlightening advices had gudided the author during all these years.


Introduction
This paper is devoted to the study of a class of integral-difference operators.The original idea of rigorous mathematical investigation of the properties of these operators is due to Professor Ilya Prigogine and goes back to 1997.The reason is that such operators appear in nonequilibrium statistical physics models [11] and became a subject of interest for physicists [10].At the same time, these operators have interesting and delicate mathematical properties, so "intuitive" physical approach may not work [10].
The operators under consideration have the form acting in the Hilbert space L 2 (R,dx).Here ϕ(x) is the so-called equilibrium distribution function, having the following properties induced by its physical nature as a probability distribution: As shown in our previous papers [5,6,7,8,9], spectral properties of ϕ depend essentially on the properties of equilibrium distribution function ϕ(x).In particular, there is a very important distinction between the cases when ϕ(x) has and does not have compact support.In the first case, it is also important to know if there exists such ε > 0 that ϕ(x) ≥ ε for all x ∈ supp ϕ.
The paper is organized as follows.In Section 2, we present some general properties of the family of operators ϕ .In Section 3, we consider operators with equilibrium distribution function having compact support.We introduce an appropriate reference operator 0 , develop complete spectral analysis for 0 , and obtain results for ϕ using the resolvent comparison approach.Spectral estimations for the eigenvalues of ϕ are presented.We also discuss the contribution of the complement to the support of ϕ(x) to the spectrum of ϕ .In Section 4, we discuss one of the most physically interesting cases of Gaussian equilibrium distribution function ϕ(x) and demonstrate that spectral properties of the corresponding collision operator ϕ differ drastically from the case of equilibrium distribution functions with a compact support.Section 5 contains the discussion of the results and of open problems.

Fourier transform, adjoint operator, and selfadjointness in weighted space
We note that one cannot represent ϕ as a difference of two operators corresponding to two terms in the nominator, as the two corresponding integrals will not converge.Another important point is that ϕ is not an integral operator as it is impossible to construct the corresponding integral kernel.However, it happens that under some simple conditions on ϕ(x), Fourier transform of ϕ is an integral operator.We use Fourier representation in order to prove some important properties of operators ϕ .
Lemma 2.1.For any real-valued function ϕ(x) ∈ L 2 (R) ∩ L 1 (R), the operator ϕ acting in the space L 2 (R) obeys the relation where * ϕ is the adjoint operator and ϕ stands for the operator of multiplication by the function ϕ(x).
Proof.The proof [5,6] is based on Fourier transform defined in a standard way: Consider the operator 3) It is an integral operator with the kernel (2.4) Yuri B. Melnikov 223 Above we used the change of variables t = s − x.As ϕ(x) ∈ L 2 (R) ∩ L 1 (R), we can interchange the integrals and obtain |t| dt, (2.5) where ϕ(k (2.6) The latter integral is known [4] to be (2.7) Therefore, we have The kernel of the adjoint operator * ϕ can be obtained from the latter expression by the interchange of the arguments k, k and complex conjugation: (2.9) Here we have used the condition ϕ(x 224 Spectral analysis for integral-difference operators The kernel of the operator F( ϕ • ϕ) is given by the convolution (2.10) The lemma is proved.
Note that ϕ is not an integral operator, but its Fourier transform ϕ is an integral operator with a weakly singular kernel.
From Lemma 2.1, we obviously have the following corollary.

Equilibrium distribution functions with compact support
In this section, we consider an important case supp ϕ ⊆

3.1.
Reference operator 0 : introduction.In this subsection, we introduce reference operator 0 as a prototype for operators ϕ corresponding to ϕ(x) with compact support suppϕ ⊆ [a,b].Operator 0 corresponds to the equilibrium distribution function Note that in order to satisfy the normalization condition 1] , but for technical convenience, we will omit the coefficient 1/2.
Operator 0 allows complete spectral analysis [5,6], which is based on the decomposition of the Hilbert space L 2 (R) into the orthogonal sum of the subspaces It corresponds to the representation of an arbitrary function u ∈ L 2 (R) in the form where u 0 = P 0 u; u ± = P ± u; P − , P 0 , and P + stand for the projection operators on the subspaces L 2 (−∞,−1), L 2 [−1,1], and L 2 (1,∞), correspondingly (operators of multiplication by the indicators of the corresponding intervals).In this representation, operator 0 can be written in the form P − 0 P + P − 0 P 0 P − 0 P + P 0 0 P + P 0 0 P 0 P 0 0 P + P + 0 P + P + 0 P 0 By straightforward calculations [5,6], one can check that the matrix elements of the above operator-valued matrix are as follows.

Study of the restricted operator
Theorem 3.2.The operator K 0 in the Hilbert space L 2 [−1,1] is selfadjoint, K 0 = K * 0 , and its spectrum σ(K 0 ) is discrete and equal to the set of simple eigenvalues where The correspondent eigenfunctions are Legendre polynomials: Proof.Selfadjointness of the operator K 0 follows from Lemma 2.1.Indeed, in our case, ϕ(x) = χ [−1,1] (x), and due to Lemma 2.1, we have 0 The operator of multiplication by the indicator χ [−1,1] (x) is the projection operator P 0 , therefore, P 0 = P 0 * , which implies Now we consider a polynomial of the order n: and demonstrate that we can uniquely choose the coefficients b (n) k such that b (n) n = 0 and K 0 p n = µ n p n .One can represent (3.15) Therefore, (3.16) We introduce the notation and rewrite the latter equality as follows: (3.18) In order to satisfy the equation K 0 p n = µp n , we have to make equal the coefficients at all the powers of x: This (n + 1)-dimensional system is equivalent to the vector equation where M = diag{µ 0 ,µ 1 ,µ 2 ,...,µ n } and A is an upper-triangular matrix with zero main diagonal (a nilpotent matrix) with the matrix elements A k j = α j−k , j ≥ k + 1: One can notice that due to the definition of the coefficients α j , they vanish for all odd indices j.The determinant D µ of the matrix (A + M − µI) is equal to hence the eigenvalues are µ = 0,µ 1 ,µ 2 ,...,µ n .The only eigenvector with b (n) n = 0 (which implies that the polynomial p n (x) is of the order n) corresponds to µ = µ n .Due to the triangular form of (3.20), all the coefficients b (n) j can be constructed by a recurrent procedure and they determine the polynomial p n (x) ∈ L 2 [−1,1] satisfying the equation K a p n (x) = µ n p n (x).Therefore, the set {µ n } ∞ n=0 is in the discrete spectrum of the operator K 0 and the correspondent eigenfunctions are the polynomials p n (x).As K 0 = K * 0 , polynomials p n (x) are Legendre polynomials and σ(K 0 ) = {µ n } ∞ n=0 .
3.3.Spectrum of the reference operator 0 .Now we are ready to describe [5,6] the spectrum of the reference operator 0 .Using representation (3.9), one can see that the spectral problem is equivalent to the following system of equations: Together with Theorem 3.2, it allows [5,6] to get the following result.
n=0 of the restricted operator K 0 is the discrete spectrum of the operator 0 and the correspondent eigenfunctions are χ , then point µ has double multiplicity and the correspondent generalized eigenfunctions have the form where Yuri B. Melnikov 229 polynomials p n (x) are the normalized eigenfunctions of the operator K 0 (Legendre polynomials), and

.29)
Proof.First, we notice that the spectrum σ(K 0 ) of the restricted operator K a is a subset of the spectrum σ() of the operator .Indeed, one can assume u − ≡ 0, u + ≡ 0; then system (3.26)turns into the equation K 0 u 0 = µu 0 and an eigenfunction u µ 0 (x) of the operator K 0 generates an eigenfunction χ [−1,1] (x)u µ 0 (x) of the operator 0 .Now, we assume that a positive value µ / ∈ σ(K 0 ).Then at least one of the functions u − , u + should not be identical to the zero.The first and the third equations of system (3.26)imply that µ should belong to the image of the function q − (x) and/or to the image of the function q + (x).One can check that the positive semiaxis R + is the image of both functions q − and q + .We assume that u − (x) ≡ 0; then u µ + (x) is a delta function with the support in the point λ + µ such that q + (λ + µ ) = µ > 0. One can check that hence Using statement (3) of Lemma 3.1 and (3.31), we get Under the assumption that µ / ∈ σ(K 0 ), the operator (K 0 − µ) is invertible and using (3.26) and (3.32), we have The similar result is true if we assume that u + ≡ 0. Then for the function u µ − (x), we have As proved in Theorem 3.2, the operator K 0 is selfadjoint, therefore, its resolvent can be represented as follows: Therefore, The legendre polynomials p n (x) are the eigenfunctions of the restricted operator K 0 , therefore, they satisfy the equality . We take x = λ ± µ in the latter equality and get (3.39) Using (3.36) and (3.39), we obtain We have shown that both the spectrum σ(K 0 ) of the restricted operator K 0 and its complement R + \σ(K 0 ) are subsets of the spectrum σ( 0 ), which means R + ⊆ σ( 0 ).On the other hand, the spectrum of the operator K 0 is nonnegative.Therefore, if µ / ∈ R + and µ ∈ σ( 0 ), either the first or the third equations of the system (3.26)should be satisfied with nontrivial u − or u + .However, that is impossible, because the images of the functions q ∓ are positive.Therefore, σ( 0 ) ⊆ R + and finally we have σ( 0 ) = R + .

Yuri B. Melnikov 231
The corresponding generalized eigenfunctions are where y ± λ are the inverse images of the function Condition (iii) is not very restrictive from a physical point of view.Indeed, smoothness of the equilibrium distribution function is a natural physical property.As the absolute values of the parameters a and b can be arbitrary large, one could think that condition (i) is also not physically restrictive.However, as we will show later, condition (ii) and related condition (i) are crucial for the spectral properties of the corresponding operators.
Proof of Theorem 3.4.Using the technique used above for the analysis of the operator 0 , we decompose the Hilbert space L 2 (R) in the orthogonal sum of the subspaces: which corresponds to the representation of an arbitrary function u ∈ L 2 (R) in the form where u ∓ = P ∓ u, u ab = P ab u.In this representation, the operator ϕ can be written in the form By straightforward calculations similar to the ones in Lemma 3.1 using condition (i) of our theorem, one can check the following: (1) P − ϕ P ab = P − ϕ P + = P + ϕ P ab = P + ϕ P − = 0; (2) P − ϕ P − and P + ϕ P + are the operators of multiplication by the functions χ − (x)q ϕ (x) and χ + (x)q ϕ (x), respectively, where the function q ϕ (x) is as follows: where 232 Spectral analysis for integral-difference operators (4) the restricted operator P ab ϕ P ab = K ϕ acts in the space L 2 [a,b] as follows: Therefore, one can write the operator ϕ in the form similar to (3.9): where q ϕ stands for the operator of multiplication by q ϕ (x).Hence, the spectral problem becomes equivalent to the system of equations similar to (3.26): (3.50)There always exist trivial solutions of the first and the third equations of system (3.50),u − (x) ≡ 0, u + (x) ≡ 0. In this case, the second equation of this system turns into K ϕ u ab = λu ab , and every eigenfunction u λ ab (x) of the restricted operator K ϕ generates an eigenfunction u λ (x) = χ ab (x)u λ ab (x) of the operator ϕ .Therefore, the spectrum of the operator K ϕ is a subset of the spectrum of the operator ϕ .
Similar to Theorem 3.3, one can show that every point λ ∈ R + belongs to the spectrum of the operator ϕ , and if λ ∈ R + \σ(K ϕ ), it has a double multiplicity.The corresponding generalized eigenfunctions can be also calculated similar to Theorem 3.3.
In order to accomplish the proof of our theorem, we have to study the spectrum σ(K ϕ ) of the operator K ϕ .As shown above, σ(K ϕ ) ⊂ σ( ϕ ).
Due to Lemma 2.1, under our conditions, the operator ϕ obeys the relation ϕ , where * ϕ is the adjoint operator and ϕ stands for the operator of multiplication by the function ϕ(x).Therefore, due to condition (i), we have Hence, due to condition (ii), the operator K ϕ is selfadjoint in the space L 2 ([a,b],dx/ϕ(x)).Consequently, its spectrum is real.
As mentioned above, by linear change of variables, without losing generality, we can assume that a = −1, b = 1.Hereafter in this proof, we accept this assumption.
By straightforward calculation, one can get where (K 0 ϕ) is the operator of multiplication by the function (K 0 ϕ)(x).Indeed, (3.52) We denote by R ϕ (z) = (K ϕ − z) −1 the resolvent of the operator K ϕ .One can check that the following relation is valid: if the inverse operator in the right-hand side of (3.54) exists.Now, we consider the resolvent R 0 (z) = (K 0 − z) −1 of the operator K 0 .Obviously, this resolvent has a discrete spectrum with the eigenvalues γ n (z) = 1/(µ n − z).For every z / ∈ σ(K 0 ), the set {γ n (z)} is bounded from above, |γ n (z)| ≤ 1/ min n |µ n − z|, and has one accumulation point γ ∞ = 0, that is, γ n → 0 when n → ∞.Therefore, except for the discrete countable set z = µ n , the resolvent R 0 (z) is a compact operator.
Due to condition (ii) of our theorem, the operators of multiplication by the functions ϕ(x) and 1/ϕ(x) are bounded.Due to condition (iii), the same is true for the operator of multiplication by the function (K 0 ϕ)(x).Indeed, Therefore, as the resolvent R 0 (z) is a compact operator except for a countable discrete set of z, the operator [z(ϕ − I) − (K 0 ϕ)] • R 0 (z) • (1/ϕ) is compact outside of this set because it is a product of a compact and a bounded operator.Hence, his spectrum cannot have an accumulation point at λ = −1.Outside of the mentioned set, it is an analytic operator-valued function of z.Consequently, the point λ = −1 can be an eigenvalue of this operator only for a countable discrete set of z.Therefore, outside of this set, there exists a bounded operator Then in the right-hand side of (3.54), we have a product of a bounded operator and a compact operator R 0 (z).Therefore, the resolvent R ϕ (z) is a compact operator except for a countable discrete set of z.This implies that the operator K ϕ can have only a discrete spectrum and its eigenvalues τ n → ∞ when n → ∞.These eigenvalues form the discrete spectrum of the operator ϕ .
3.5.Spectral estimation for the discrete spectrum.As shown above, in the case suppϕ = [a,b], the spectrum of the restricted operator K ϕ = P ab ϕ P ab generates discrete spectrum of the operator ϕ .Without losing generality, we can assume that a = −1 and b = 1.We will use representation (3.51) in order to obtain spectral estimations for operator K ϕ .We assume that ϕ(x) satisfies the conditions of Theorem 3.4.
As shown above, operator K ϕ is selfadjoint in Hilbert space L 2 ([−1,1],dx/ϕ(x)).We denote the scalar product in this space as follows: and the norm as IuI 2 := u,u .Operator K 0 is selfadjoint in Hilbert space L 2 ([−1,1],dx); we denote the scalar product in this space as follows: and the norm as u 2 := (u,u).We consider the spectral problem As shown above, under the conditions of Theorem 3.4, operator K ϕ is semibounded from below, has purely discrete spectrum, and selfadjoint in Hilbert space L 2 ([−1,1],dx/ϕ(x)).Due to the maximinimal principle [1], the nth eigenvalue of operator K ϕ can be calculated as follows: where D ϕ is the domain of operator K ϕ and Φ n is an arbitrary linear set satisfying condition dimD ϕ \Φ n ≤ n.Using representation (3.51), we get from (3.59) Obviously, As ϕ(x) > 0, one can easily check that min Operator K 0 is selfadjoint semibounded from below, has purely discrete spectrum in Hilbert space L 2 ([−1,1],dx).Using the maximinimal principle [1], we get max where µ n , defined in Theorem 3.2, are the eigenvalues of operator K 0 .Under the conditions of Theorem 3.4, the domains D ϕ and D 0 of the operators K ϕ and K 0 coincide: D ϕ = D 0 .Therefore, combining (3.62), (3.65), and (3.66), we obtain the following estimation for the eigenvalues of operator K ϕ : (3.67) In particular, this estimation can be applied to the case when the equilibrium distribution function ϕ(x) is similar to the homogeneous equilibrium distribution function where γ ∈ R, In this case, one can easily estimate max

.70)
In this case, (3.68) can be written as follows: We say that ϕ(x) is similar to the homogeneous equilibrium distribution function if it has the form (3.68) and γ α/2A, γ 1/C.In this case, it is easy to construct approximations of the eigenvalues and eigenfunctions of the operator K ϕ in terms of the powers of the small parameter γ.We will write down here the first-order approximation.From (3.51), (3.58), and (3.68), we have Decomposing the eigenfunction u in a series of normalized Legendre polynomials u = m≥0 u (m) p m (x) and calculating the scalar product of the both hand sides of (3.72) with p l (x), we get From this equation, we get the following first-order approximations for nth eigenvalue and eigenfunction of operator K 0 : Cµ n , and γ α/2A.

Continuous spectrum generated by the complement to the support of the equilibrium distribution function.
As one can see from the previous discussion, the support of the equilibrium distribution function ϕ(x) is "responsible" for the discrete spectrum of the operator ϕ (coinciding with the spectrum of the restricted operator K ϕ = P supp ϕ ϕ P suppϕ ), while the complement to the support "generates" branches of the continuous spectrum given by the image of the function q ϕ (x) when x / ∈ suppϕ.We will show that every interval, where equilibrium distribution function ϕ(x) vanishes, generates a branch of continuous spectrum of the corresponding operator ϕ .Different cases are illustrated with several specific examples.In order to be able to consider in a unified manner several specific cases, we generalize our original problem for spaces L 2 (I), where I ⊆ R is an interval on the real line (it may coincide with R as above).We define operators ϕ as follows: Above we have shown (Theorem 3.4) that if I = R, ϕ(x) has a compact support suppϕ, and ϕ(x) is smooth and separated from zero in suppϕ, then the complement to suppϕ generates branch [0,∞) of absolutely continuous spectrum of double multiplicity of operator ϕ .Below we demonstrate that every separated interval belonging to the complement of the support of equilibrium distribution function ϕ(x) generates a branch (finite or infinite) of the spectrum of operator ϕ .Let I = [A,B] ⊆ R be an interval on the real line R (may be coinciding with R).Let D N = j [a j ,b j ], a j < b j < a j+1 < b j+1 , be a union (finite or infinite) of intervals, where equilibrium distribution function ϕ(x) vanishes: ϕ(x)| DN ≡ 0. We denote by D S = I\D N the complement to the set D N and assume that ϕ(x) is piecewise continuous on D S .We also use standard assumptions ϕ(x) ≥ 0 and I ϕ(x)dx = DS ϕ(x)dx = 1.Following the scheme introduced above, we represent Hilbert space L 2 (I) as L 2 (I) = L 2 (D S ) ⊕ L 2 (D N ) and denote by P S , P N the projection operators on subspaces L 2 (D S ), L 2 (D N ), respectively.By χ S (x), χ N (x), we denote the indicators of the sets D S , D N , respectively.Any function u ∈ L 2 (I) can be presented as follows: 238 Spectral analysis for integral-difference operators In this representation, operator ϕ can be written as follows: and one can calculate the entities of this operator matrix: where Therefore, operator ϕ in representation (3.77) can be written as follows: where q stands for the operator of multiplication by function q(x) defined in the set D N .Hence, the spectral problem ϕ u = λu is equivalent to the system of equations This immediately implies the following lemma.
Proof.Suppose that λ ∈ (q), that is, there exists at least one x λ ∈ D N such that q(x λ ) = λ.We choose u N (x) = δ(x − x λ ), so the second equation of system (3.81) is satisfied.We calculate . First, we assume that λ / ∈ σ(K S S); therefore, there exists bounded operator (K SS − λ) −1 , and from the first equation of system (3.81),we calculate If λ ∈ σ(K SS ), that is, there exists (generalized) eigenfunction u λ S such that K SS u λ S = λu λ S , we take and system (3.81) is again satisfied.
Therefore, image (q) of function q(x), x ∈ D N = j [a j ,b j ], generates part of the spectrum of operator ϕ .It is worth noticing that as ϕ(x) ≥ 0, also q(x) ≥ 0, that is, (q) ⊆ R + .
We consider (q) for every interval [a j ,b j ] ⊂ D N .For x ∈ [a j ,b j ], we can write Values ν ± j depend on the behaviour of function ϕ(x) in the points x = a j − 0 and x = b j + 0; they can be either finite or infinite.Now, we illustrate different cases with three specific examples.
Example 3.6 (absolutely continuous spectrum of infinite multiplicity).We consider here case I = R and infinite number of intervals [a j ,b j ] ⊂ D N .Namely, we define equilibrium distribution function ϕ(x) as follows: One can easily calculate Note that in this case, ϕ(x) does not have a compact support.
We consider interval [2 j − 1,2 j], j > 1, where function ϕ(x) vanishes.For x ∈ [2 j − 1,2 j], function q(x) can be represented as a sum of the principal part q p j (x) (given by the integrals over neighbouring intervals from D S ) and the remaining part qj (x): (3.87) 240 Spectral analysis for integral-difference operators Function q p j (x) goes to infinity at x = 2 j − 1 and x = 2 j and has the only minimum in the interval x ∈ [2 j − 1,2 j] at the point The remaining part given by the integrals over the rest of the integrals can be easily estimated as with some constants C 1 and C 2 .Therefore, function q(x) maps interval [2j − 1,2 j] into semiinfinite interval [M j ,∞), where 0 < M j = O(| j| −1 ) as j → ∞.The same is true for j < 0. It means that open interval (0,∞) belongs to the absolute continuous spectrum of operator ϕ and has infinite spectral multiplicity.
Example 3.7 (branch of absolutely continuous spectrum with lower boundary parametrically going to infinity).Now, we consider the case I = [−1,1] with equilibrium distribution function defined as follows: x ∈ (−a,a). (3.89) One can calculate and see that in the interval [−a,a], function q(x) → ∞ when x → ±a and has the only minimum at the point ) is a branch of absolutely continuous spectrum of operator ϕ , and its lower boundary goes to infinity when the complement to the support of function ϕ(x) shrinks, that is, when a → 0.
Example 3.8 (finite branch of absolutely continuous spectrum).Finally, we consider a case of continuous equilibrium distribution function.Namely, let I = R and

Gaussian equilibrium distributions
Contrary to most of the above-considered cases, physically important Gaussian equilibrium distributions [8,9,10] have infinite support and the corresponding operators cannot be analyzed using the above described technique.We consider a family of operators K ϕa with truncated Gaussian equilibrium distribution function on the interval [−a,a].Without loss of generality, one can take β = 1.Infinite integration limits in the original operator (1.1) are always understood as the limit of the integral over the interval [−a,a] when a → ∞.One can have an intuitive feeling [10] that, as truncated Gaussian functions are "very similar" for different but large values of the truncation parameter a, the spectral properties of the corresponding operators ϕa will also be similar for different large values of a.However, that is not true.Namely, there is no regular limit of the operator ϕa at a → ∞.Therefore, there is no way to develop a successful perturbation theory for the spectrum of the operator ϕa with respect to the parameter 1/a.Previously [9], we have proved analytically that the first two eigenvalues λ 1 , λ 2 of operator ϕa go to zero ∼ a −1 when a → ∞ and have confirmed it numerically for several other lower eigenvalues.A stronger result [8] is an analytic proof of the fact that zero becomes a point of spectral concentration when a → ∞, that is, the number of the eigenvalues in arbitrary small vicinity of zero increases unlimitedly as a → ∞.
As the discrete spectrum of the original operator ϕ is determined by the spectrum of the restricted operator K ϕ , in this section, we study a family of operators on the interval [−a,a] with the equilibrium distribution function ϕ(x) given by (4.1) with β = 1.One can see that a simple change of variables makes the spectral problem for operator K ϕa equivalent to the spectral problem for operator Indeed, introducing the notations s = s/a, x = x/a, and ũ(x) = u(ax), one can calculate It is more convenient to study the spectral properties of our operator in the form (4.3).In this form, it is obvious that functions ϕ a (x) given by (4.4) are not "similar" for different but large values of the parameter a.
We use notations for the inner product in space L 2 ([−1,1],dx/ϕ a (x)) and for the inner product in the space L 2 ([−1,1],dx).We denote by E a [−M,M] the spectral measure of the operator K a on the interval [−M,M] ⊂ R. By Ᏼ a = L 2 ([−1,1],dx/ϕ a (x)), we denote Hilbert space where operator K a acts as a selfadjoint operator.The main result [8] presented in this section is the following theorem.
This theorem means that the number of eigenvalues (counted with multiplicity) of the operator K a (and, consequently, of the operator K ϕa ) in arbitrary small vicinity of zero increases to infinity when the truncation parameter a goes to infinity.Indeed, due to Theorem 3.4, the spectrum of the operators K ϕa is purely discrete for all a < ∞.Hence the increase of the spectral measure on the interval [−M,M] can be caused only by the increase of the number of the eigenvalues (counted with multiplicity) on this interval.Therefore, zero is a point of spectral concentration for the limit operator ∞ = lim a→∞ K ϕa .
Proof of Theorem 4.1.We will prove this theorem using the bilinear form approach.In order to prove our theorem, it is enough [1,2] to construct for all N > 0 a linear set F a N ⊂ D(K a ), dimF a N = N, such that for any M > 0, there exists a 0 (N,M) such that for all Yuri B. Melnikov 243 a > a 0 (N,M), inequality is true for all u ∈ F a N .We construct F a N as a linear span: On the order hand, using representation (3.51), we have We first estimate the term with a simple change of variables between x and s in the second term.We estimate the terms (ϕ a ,K 0 (p k p l )).As p k , p l ∈ F a N , then k,l ≤ N − 1; therefore, the product of these Legendre polynomials is a polynomial of the power not higher than 2N − 2. Therefore, one can represent where for all k, l, m.Therefore, Using the Laplace method [3], we find the asymptotics (4.17) Obviously, C a < C 1 for a > 1.Thus we got the estimate at a → ∞: Obviously, Finally, we have obtained the estimate does not depend on a and finite for any N < ∞.Now, we estimate the term

.23)
We have

Discussion and open problems
In the present paper, we have summarized the known results on the spectral analysis of a class of integral-difference operators ϕ for different classes of equilibrium distribution functions ϕ(x).The most advanced results can be obtained in the case when equilibrium distribution function has compact support and uniformly separated from zero in the whole support.However, even in this case, there is no example of exactly solvable spectral problem except for the reference operator 0 given by (3.2).In the case of compact support, the spectral analysis of the original operator ϕ is essentially reduced to the analysis of the restricted operator K ϕ = P supp ϕ ϕ P supp ϕ .For the reference operator 0 , it is shown that the corresponding restricted operator K 0 commutes with second-order differential operator L generating Legendre polynomials.It would be rather interesting to find such differential or pseudodifferential operator that commutes with K ϕ for some other ϕ.
[a,b] ⊂ R. Without losing generality, one can assume that a = −1, b = 1.Indeed, it is enough to make linear change of variables x → x = (2x − b − a)/(b − a) to get equivalent problem on the renormalized real line such that [a,b] → [−1,1].
ϕ corresponding to equilibrium distribution functions ϕ(x) with compact support.Let suppϕ(x) = [a,b].We denote by P ab the operator of multiplication by the indicator of the interval [a,b], and by P − and P + the operators of multiplication by the indicators of the intervals (−∞,a) and (b,∞), correspondingly.The main result can be formulated as follows [7].Theorem 3.4.Let the equilibrium distribution function ϕ(x) satisfy the following conditions: (i) ϕ(x) has a compact support: suppϕ(x) ⊆ [a,b]; (ii) ϕ(x) is bounded, positive, and separated from zero on [a,b] :