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In this paper a solution of the direct Cauchy problems for heat equation is founded in the Hermite polynomial series form. A well-known classical solution of direct problem is represented in the Poisson integral form. The author shows the…
It has long been known that there exists a coordinate transformation which exactly maps the quantum free particle to the quantum harmonic oscillator. Here we extend this result by reformulating it as a unitary operation followed by a time…
We discuss propagation of space-time singularities for the quantum harmonic oscillator with time-dependent metric and potential perturbations. Reformulating the quasi-homogeneous wave front set according to Lascar (1977) in a semiclassical…
We consider fractional relaxation and fractional oscillation equations involving Erdelyi-Kober integrals. In terms of Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients.…
The inverted harmonic oscillator problem is investigated quantum mechanically. The exact wave function for the confined inverted oscillator is obtained and it is shown that the associated energy eigenvalues are discrete and it is given as a…
A nonlinear model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature is exactly solved. This model depends of a parameter $\la$ that is related with the curvature of the space. Firstly the relation with other…
We transform the time-dependent Schroedinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is…
We introduce a new class of piece-wise quadratic potentials for nonlinear wave equations with a kink solutions. The potentials allow an exact description of the spectral properties for the linearized equation at the kink. This description…
A selfconsistent system of equations describing evolution of linear spherically symmetric perturbations in Fridmann world by an arbitrary equation of state is obtained. The perturbations singular part corresponding to massive particle-like…
This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their…
The classical and quantum dynamics of simple time-reparametrization- invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description…
We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted H\"older and Sobolev spaces with discrete…
In this investigation, the displacement operator is revisited. We established a connection between the Hermitian version of this operator with the well-known Weyl ordering. Besides, we characterized the quantum properties of a simple…
We present a general formalism for the Hamiltonian description of perturbation theory around any spatially homogeneous spacetime. We employ and refine the Dirac method for constrained systems, which is very well-suited to cosmological…
In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical…
In this article we provide a method for establishing operator-type error estimates between solutions to rapidly oscillating evolutionary equations and their homogenised counter parts. This method is exemplified by applications to the wave,…
In this paper we study reducibility of time quasiperiodic perturbations of the quantum harmonic or anharmonic oscillator in one space dimension. We modify known algorithms obtaining a reducibility result which allows to deal with…
In this work we address the problem of the quantization of a simple harmonic oscillator that is perturbed by a time dependent force. The approach consists of removing the perturbation by a canonical change of coordinates. Since the…
In this article we propose the calculation of the unconditional Wiener measure functional integral with a term of the fourth order in the exponent by an alternative method as in the conventional perturbative approach. In contrast to the…
The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schr\"{o}dinger equation, which are determined by any two independent solutions to the classical equation of motion.…