# Vol. 20 (2014), no. 1

### Volodymyr Dmytrovych Koshmanenko (to his 70th birthday)

Methods Funct. Anal. Topology **20** (2014), no. 1, 1-2

### On large coupling convergence within trace ideals

Methods Funct. Anal. Topology **20** (2014), no. 1, 3-9

Let $\mathcal E$ and $\mathcal P$ be nonnegative quadratic forms such that $\mathcal E + b \mathcal P$ is closed and densely defined for every nonnegative real number $b$. Let $H_b$ be the selfadjoint operator associated with $\mathcal E + b\mathcal P.$ By Kato's monotone convergence theorem, there exists an operator $L$ such that $(H_b+1)^{-1}$ converges to $L$ strongly, as $b$ tends to infinity. We give a condition which is sufficient in order that the operators $(H_b+1)^{-1}$ converge w.r.t. the trace norm with convergence rate $O(1/b)$. As an application we discuss trace norm resolvent convergence of Schrodinger operators with point interactions.

### On the discrete spectrum of a linear operator pencil arising in transport theory

Methods Funct. Anal. Topology **20** (2014), no. 1, 10-16

We study the problem of the finiteness of the discrete spectrum for linear operator pencils occurring in one-velocity transport theory. The results are obtained using direct methods of perturbation theory for linear operators. The proposed approach allowed to give a relatively quick proofs of the main results improving related results obtained previously by K. M. Case and C. G. Lekkerkerker.

### Delta-type solutions for a system of induction equations with discontinuous velocity field

A. I. Esina, A. I. Shafarevich

Methods Funct. Anal. Topology **20** (2014), no. 1, 17-33

We study asymptotic solutions of a Cauchy problem for induction equations describing magnetic field in a well conducting fluid. We assume that the coefficient (the velocity field of the fluid) changes rapidly in a small vicinity of a two-dimensional surface. We prove that the weak limit of the solution has delta-type singularity on this surface; in the case of a perfectly conducting fluid, we describe several regularizations of the problem with discontinuous coefficients which allow to define generalized solutions.

### Schrödinger operators with non-symmetric zero-range potentials

Methods Funct. Anal. Topology **20** (2014), no. 1, 34-49

Non-self-adjoint Schrödinger operators $A_{\mathbf{T}}$ which correspond to non-symmetric zero-range potentials are investigated. For a given $A_{\mathbf{T}}$, a description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the possibility of interpretation of $A_{\mathbf{T}}$ as a self-adjoint operator in a Krein space is studied, the problem of similarity of $A_{\mathbf{T}}$ to a self-adjoint operator in a Hilbert space is solved.

### On a criterion of mutual adjointness for extensions of some nondensely defined operators

Methods Funct. Anal. Topology **20** (2014), no. 1, 50-58

In the paper the role of initial object is played by a pair of closed linear densely defined operators $L_0$ and $M_0$, where $L_0 \subset M_0^{\ast}:= L,$ acting in Hilbert space. A criterion of mutual adjointness for some classes of the extensions of finite-dimensional (non densely defined) restrictions of $L_0$ and $M_0$ are established. The main results are based on the theory of linear relations in Hilbert spaces and are formulated in the terms of abstract boundary operators.

### On the common point spectrum of pairs of self-adjoint extensions

Methods Funct. Anal. Topology **20** (2014), no. 1, 59-67

Given two different self-adjoint extensions of the same symmetric operator, we analyse the intersection of their point spectra. Some simple examples are provided.

### Eigenfunction expansions associated with an operator differential equation non-linearly depending on a spectral parameter

Methods Funct. Anal. Topology **20** (2014), no. 1, 68-91

For an operator differential equation that depends on a spectral parameter in the Nevanlinna manner we obtain expansions in eigenfunctions.

### Continuous dual of $c_0(Z,X,\bar\lambda, \bar p)$ and $c(Z,X,\bar\lambda, \bar p)$

Riti Agrawal, J. K. Srivastava

Methods Funct. Anal. Topology **20** (2014), no. 1, 92-100

A bilateral sequence is a function whose domain is the set $Z$ of all integers with natural ordering. In this paper we study the continuous dual of the Banach space of $X$-valued bilateral sequence spaces $c_0(Z,X,\bar\lambda, \bar p)$ and $c(Z,X,\bar\lambda, \bar p)$.