K. Górska

Squeezed states and Hermite polynomials in a complex variable

Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pure Appl. Math. 14, 187 (1961)].

The higher-order heat-type equations via signed Lévy stable and generalized Airy functions

We study the higher-order heat-type equation with first time and Mth spatial partial derivatives, M = 2, 3, .... We demonstrate that its exact solutions for M even can be constructed with the help of signed Levy stable functions. For M odd the same role is played by a special generalization of the Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spatial and temporary evolution of particular solutions for simple initial conditions.

Relativistic wave equations: an operational approach

The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schrodinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.

Symbolic methods for the evaluation of sum rules of Bessel functions

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.

On the properties of Laplace transform originating from one-sided Lévy stable laws

We consider the conventional Laplace transform of f(x), denoted by

Theory of relativistic heat polynomials and one-sided Lévy distributions

{ \mathcal L }[f(x);p]\quad \equiv \quad F(p)={\displaystyle \int }_{0}^{\infty }{{\rm{e}}}^{-{px}}f(x){\rm{d}}x

Photoluminescence decay of silicon nanocrystals and Lévy stable distributions

with

The spherical Bessel and Struve functions and operational methods

{\mathfrak{Re}}(p)\gt 0

Exact and explicit evaluation of Brézin–Hikami kernels

. For

Coherence, Squeezing and Entanglement: An Example of Peaceful Coexistence

0\lt \alpha \lt 1