We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space

Method of contour integration of the resolvent is one of the general methods of the spectral analysis of nonself-adjoint (dissipative) operators. It is related to a fine estimate of the resolvent on expanding contours which separates the spectrum. The feasibility of this method is restricted to weak perturbations of self-adjoint operators and operators having sparse discrete spectrum. Since there are no asymptotics of the solutions for a wide class of singular problems, this method cannot be applied properly.

It is well known [

In this paper, we consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space

Let

If we let

For arbitrary sequences

For any sequence

It follows from Green’s formula (

Denote by

Assume that the symmetric operator

Denote by

The

Let

For arbitrary vectors

The domain

Let us consider the following linear maps from

For any contraction

In particular, if

In what follows, we will study the dissipative operators

In order to construct a self-adjoint dilation of the maximal dissipative operator

The operator

We assume that

The self-adjoint operator

Now, it is not difficult to show that

In order to construct a self-adjoint dilation of the maximal dissipative operator

The proof of the next theorem is similar to that of Theorem

The operator

The unitary group

Property (4) is obvious. To verify property (1) for

From this, we conclude that

To prove property (2), we denote by

The scattering matrix is defined in terms of the spectral representations theory in this scheme of the Lax-Phillips scattering theory. We will continue with their construction and prove property (3) of the incoming and outgoing subspaces along the way.

We recall that the linear operator

The operator

Let

To prove property (3) we first set

The equality

Using property (1) of the subspace

Let

The

We adopt the following notations:

Let

The transformation

For

Let us extend the Parseval equality to the whole

Now, we let

The transformation

From (

Therefore, the transformation

The function

Let

Let

The characteristic function of the maximal dissipative operator

If

Let

The transformation

Let

The transformation

It is seen from (

The function

Using the explicit form of the unitary transformation

The characteristic function of the maximal dissipative operator

We know that the characteristic function of a maximal dissipative operator

Let

For all values of

It can be easily seen from (

If

The proof of the next result is similar to that of Theorem

For all values of

Since a linear operator