# S. Albeverio

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Articles: 21

### Schrödinger operators with nonlocal potentials

Methods Funct. Anal. Topology 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.

### On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols

Methods Funct. Anal. Topology 17 (2011), no. 2, 97-111

We introduce a new fine classification of singularly continuous probability measures on $R^1$ on the basis of spectral properties of such measures (topological and metric properties of the spectrum of the measure as well as local behavior of the measure on subsets of the spectrum). The theorem on the structural representation of any one-dimensional singularly continuous probability measure in the form of a convex combination of three singularly continuous probability measures of pure spectral type is proved.

We introduce into consideration and study a $\widetilde{Q}$-representation of real numbers and a family of probability measures with independent $\widetilde{Q}$-symbols. Topological, metric and fractal properties of the above mentioned probability distributions are studied in details. We also show how the methods of $\widetilde{P}-\widetilde{Q}$-measures can be effectively applied to study properties of generalized infinite Bernoulli convolutions.

### Boundary problems for the wave equation with the Lévy Laplacian in Shilov's class

Methods Funct. Anal. Topology 16 (2010), no. 3, 197-202

We present solutions to some boundary value and initial-boundary value problems for the "wave" equation with the infinite dimensional L\'evy Laplacian $\Delta _L$ $$\frac{\partial^2 U(t,x)}{\partial t^2}=\Delta_LU(t,x)$$ in the Shilov class of functions.

### Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms

Methods Funct. Anal. Topology 15 (2009), no. 2, 177-187

In the present paper derivations and $*$-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all $L^0$-linear derivations and $L^{0}$-linear $*$-automorphisms are inner. Moreover, it is proved that each $L^0$-linear automorphism of the algebra of all linear operators on a $bo$-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.

### Boundary problems for fully nonlinear parabolic equations with Lévy Laplacian

Methods Funct. Anal. Topology 14 (2008), no. 1, 1-9

We suggest a method to solve boundary and initial-boundary value problems for a class of nonlinear parabolic equations with the infinite dimensional L'evy Laplacian $\Delta _L$ $$f\Bigl(U(t,x),\frac{\partial U(t,x)}{\partial t},\Delta_LU(t,x)\Bigl)=0$$ in fundamental domains of a Hilbert space.

### Inverse spectral problems for coupled oscillating systems: reconstruction from three spectra

Methods Funct. Anal. Topology 13 (2007), no. 2, 110-123

We study an inverse spectral problem for a compound oscillating system consisting of a singular string and $N$~masses joined by springs. The operator $A$ corresponding to this system acts in $L_2(0,1)\times C^N$ and is composed of a Sturm--Liouville operator in $L_2(0,1)$ with a distributional potential and a Jacobi matrix in~$C^N$ that are coupled in a special way. We solve the problem of reconstructing the system from three spectra---namely, from the spectrum of $A$ and the spectra of its decoupled parts. A complete description of possible spectra is given.

### The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles

Methods Funct. Anal. Topology 13 (2007), no. 1, 1-16

A model operator associated with the energy operator of a system of three non conserved number of particles is considered. The essential spectrum of the operator is described by the spectrum of a family of the generalized Friedrichs model. It is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum, if a generalized Friedrichs model has a zero energy resonance.

### Lévy-Dirichlet forms. II

Methods Funct. Anal. Topology 12 (2006), no. 4, 302-314

A Dirichlet form associated with the infinite dimensional symmetrized Levy-Laplace operator is constructed. It is shown that there exists a natural connection between this form and a Markov process. This correspondence is similar to that studied in a previous paper by the same authors for the non-symmetric Levy Laplacian.

### $*$-wildness of some classes of $C^*$-algebras

Methods Funct. Anal. Topology 12 (2006), no. 4, 315-325

We consider the complexity of the representation theory of free products of $C^*$-algebras. Necessary and sufficient conditions for the free product of finite-dimensional $C^*$-algebras to be $*$-wild is presented. As a corollary we get criteria for $*$-wildness of free products of finite groups. It is proved that the free product of a non-commutative nuclear $C^*$-algebra and the algebra of continuous functions on the one-dimensional sphere is $*$-wild. This result is applied to estimate the complexity of the representation theory of certain $C^*$-algebras generated by isometries and partial isometries.

### Dynamics of discrete conflict interactions between non-annihilating opponents

Methods Funct. Anal. Topology 11 (2005), no. 4, 309-319

### On real $AW^*$-algebras

Methods Funct. Anal. Topology 11 (2005), no. 2, 99-112

### Dense subspaces in scales of Hilbert spaces

Methods Funct. Anal. Topology 11 (2005), no. 2, 156-169

### Riquier problem for nonlinear elliptic equations with Lévy Laplacians

Methods Funct. Anal. Topology 11 (2005), no. 1, 1-9

### Schrödinger operators with a number of negative eigenvalues equal to the number of point interactions

Methods Funct. Anal. Topology 9 (2003), no. 4, 273-286

### Fine structure of the singular continuous spectrum

Methods Funct. Anal. Topology 9 (2003), no. 2, 101-119

### Quasi-invariance and Gibbs structure of diffusion measures on infinite product groups

Methods Funct. Anal. Topology 6 (2000), no. 1, 28-42

### Generalized eigenfunctions under singular perturbation

Methods Funct. Anal. Topology 5 (1999), no. 1, 13-28

### Some examples of Dirichlet operators associated with the actions of infinite dimensional Lie groups

Methods Funct. Anal. Topology 4 (1998), no. 2, 1-15

### Lippmann-Schwinger equation in the singular perturbation theory

Methods Funct. Anal. Topology 3 (1997), no. 1, 1-27

### Quantum hierarchical model

Methods Funct. Anal. Topology 2 (1996), no. 3, 1-35

### Dirichlet operators semigroups in some Gibbs lattice spin systems

Methods Funct. Anal. Topology 1 (1995), no. 1, 3-27