# Volume 52, № 5, 2000

### Yurii Makarovich Berezanskii

Gorbachuk M. L., Mitropolskiy Yu. A., Samoilenko A. M., Skrypnik I. V.

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 579-581

### Approximation of general zero-range potentials

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 582-589

A norm resolvent convergence result is proved for approximations of general Schrodinger operators with zero-range potentials. An approximation of the δ’-interaction by nonlocal non-Hermitian potentials (without a renormalization of the coupling constant) is also constructed.

### Some problems for multidimensional integro-differential equations of hyperbolic type

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 590-595

We prove the well-posedness of the Cauchy, Goursat, and Darboux problems for multidimensional in-tegro-differential equations of the hyperbolic type encountered in biology.

### On analytic solutions of operator differential equations

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 596-607

We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions of a differential equation *y*′(*t*) = *Ay*(*t*), *t* ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya.

### On one generalization of the Berezanskii evolution criterion for the self-adjointness of operators

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 608-615

We describe all weak solutions of a first-order differential equation in a Banach space on (0, ∞) and investigate their behavior in the neighborhood of zero. We use the results obtained to establish necessary and sufficient conditions for the essential maximal dissipativity of a dissipative operator in a Hilbert space.

### Averaging of integral functionals related to domains of frame-type periodic structure with thin channels

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 616-625

We establish the Γ-convergence of a sequence of integral functionals related to domains of frame-type periodic structure with thin channels. We obtain a representation for the integrand of a Γ-limit functional.

### Regularized approximations of singular perturbations from the $H_2$-class

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 626-637

For a sequence of singular perturbations belonging to the $H_1$-class and converging to a given singular perturbation from the $H_2$-class, we find a method of additive regularization that guarantees the strong resolvent convergence of perturbed operators.

### On the inverse problem for perturbations of an abstract wave equation in the Lax-Phillips scattering scheme

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 638-647

The inverse scattering problem for perturbations of an abstract wave equation is investigated within the framework of the Lax-Phillips scattering scheme.

### Existence of solutions of abstract volterra equations in a banach space and its subsets

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 648-657

We consider a criterion and sufficient conditions for the existence of a solution of the equation $$Z_t x = \frac{{t^{n - 1} x}}{{\left( {n - 1} \right)!}} + \int\limits_0^t {a\left( {t - s} \right)AZ_s xds} $$ in a Banach space *X*. We determine a resolvent of the Volterra equation by differentiating the considered solution on subsets of *X*. We consider the notion of "incomplete" resolvent and its properties. We also weaken the Priiss conditions on the smoothness of the kernel *a* in the case where A generates a *C* _{0}-semigroup and the resolvent is considered on *D(A)*.

### Lie-Kac bigroups

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 658-666

We define the Lie-Kac bigroups as special double Hilbert algebras canonically associated with ring groups (the Kac algebras) and related to the Lie bialgebras.

### Spectrum and states of the bcs hamiltonian in a finite domain. I. Spectrum

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 667-689

The BCS Hamiltonian in a finite cube with periodic boundary condition is considered. The special subspace of pairs of particles with opposite momenta and spin is introduced. It is proved that, in this subspace, the spectrum of the BCS Hamiltonian is defined exactly for one pair, and for n pairs the spectrum is defined through the eigenvalues of one pair and a term that tends to zero as the volume of the cube tends to infinity. On the subspace of pairs, the BCS Hamiltonian can be represented as a sum of two operators. One of them describes the spectra of noninteracting pairs and the other one describes the interaction between pairs that tends to zero as the volume of the cube tends to infinity. It is proved that, on the subspace of pairs, as the volume of the cube tends to infinity, the BCS Hamiltonian tends to the approximating Hamiltonian, which is a quadratic form with respect to the operators of creation and annihilation.

### Notes on infinite-dimensional nonlinear parabolic equations

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 690-701

We present a method for the solution of the Cauchy problem for three broad classes of nonlinear parabolic equations $$\frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {\Delta _L U\left( {t,x} \right)} \right), \frac{{\partial U\left( {t,x} \right)}}{{\partial t}} f\left( {t,\Delta _L U\left( {t,x} \right)} \right),$$ and $$\frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {U\left( {t,x} \right), \Delta _L U\left( {t,x} \right)} \right)$$ with the infinite-dimensional Laplacian Δ_{L}.

### On the rate of convergence of the Ritz method for ordinary differential equations

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 702-706

We obtain direct and inverse theorems on the approximation of solutions of self-adjoint boundary-value problems for the Sturm-Liouville equation on a finite interval by the Ritz method.

### On some boundary-value problems with a shift on characteristics for a mixed equation of hyperbolic-parabolic type

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 707-716

We prove theorems on the existence and uniqueness of solutions of nonlocal boundary-value problems with shift for mixed second- and third-order equations of hyperbolic-parabolic type.

### On the spectral theory of a Generalized Differential Krein System

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 717-721

We consider the generalized differential Krein system. We establish the relationship between the behavior of a solution of the system and the character of the corresponding spectral matrix function.