Vol. 19 (2013), no. 3

Fedor Semenovich Rofe-Beketov (to his 80th birthday)

Editorial Board

Article (.pdf)

MFAT 19 (2013), no. 3, 197-198


Schrödinger operators with nonlocal potentials

Sergio Albeverio, Leonid Nizhnik

↓ Abstract   |   Article (.pdf)

MFAT 19 (2013), no. 3, 199-210


We describe selfadjoint nonlocal boundary-value conditions for new exact solvable models of Schrödinger operators with nonlocal potentials. We also solve the direct and the inverse spectral problems on a bounded line segment, as well as the scattering problem on the whole axis for first order operators with a nonlocal potential.

Factorizations of nonnegative symmetric operators

Yury Arlinskiĭ, Yury Kovalev

↓ Abstract   |   Article (.pdf)

MFAT 19 (2013), no. 3, 211-226


We prove that each closed denselydefined and nonnegative symmetric operator $\dot A$ having disjointnonnegative self-adjoint extensions admits infinitely manyfactorizations of the form $\dot A=\mathcal L\mathcal L_0$, where $\mathcal L_0$ is aclosed nonnegative symmetric operator and $\mathcal L$ its nonnegativeself-adjoint extension. The same factorizations are also establishedfor a non-densely defined nonnegative closed symmetric operator withinfinite deficiency indices while for operator with finitedeficiency indices we prove impossibility of such a kindfactorization. A construction of pairs $\mathcal L_0\subset\mathcal L$ ($\mathcal L_0$ isclosed and densely defined, $\mathcal L=\mathcal L^*\ge 0$) having the property${\rm dom\,}(\mathcal L\mathcal L_0)=\{0\}$ (and, in particular, ${\rm dom\,}(\mathcal L^2_0)=\{0\}$) is given.

On square root domains for non-self-adjoint Sturm-Liouville operators

Fritz Gesztesy, Steve Hofmann, Roger Nichols

↓ Abstract   |   Article (.pdf)

MFAT 19 (2013), no. 3, 227-259


We determine square root domains for non-self-adjoint Sturm-Liouville operators of the type $$ L_{p,q,r,s} = - \frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$, where either $(c,d)$ coincides with the real line $\mathbb R$, the half-line $(a,\infty)$, $a \in \mathbb R$, or with the bounded interval $(a,b) \subset \mathbb R$, under very general conditions on the coefficients $q, r, s$. We treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case, and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval context. (In the particular case $p=1$ a.e. on $(a,b)$, we treat all separated boundary conditions at $a, b$.)

Spectral singularities of differential operator with triangular matrix coefficients

A. M. Kholkin

↓ Abstract   |   Article (.pdf)

MFAT 19 (2013), no. 3, 260-267


For a non-selfadjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, we construct an example of such an operator having spectral singularities.

On exit space extensions of symmetric operators with applications to first order symmetric systems

V. I. Mogilevskii

↓ Abstract   |   Article (.pdf)

MFAT 19 (2013), no. 3, 268-292


Let $A$ be a symmetric linear relation with arbitrary deficiency indices. By using the conceptof the boundary triplet we describe exit space self-adjointextensions $\widetilde A^\tau$ of $A$ in terms of a boundary parameter $\tau$. We characterize certain geometrical properties of $\widetilde A^\tau$ and describe all $\widetilde A^\tau$ with ${\rm mul}\, \widetilde A^\tau=\{0\}$. Applying these results to general (possibly non-Hamiltonian) symmetric systems $Jy'- B(t)y=\Delta(t)y, \; t \in [a,b\rangle,$ we describe all matrix spectral functions of theminimally possible dimension such that the Parseval equality holdsfor any function $f\in L_\Delta^2([a,b \rangle)$.

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