Related papers: Continuous vs. discrete models for the quantum har…
We propose the assumption of quantum mechanics on a discrete space and time, which implies the modification of mathematical expressions for some postulates of quantum mechanics. In particular we have a Hilbert space where the vectors are…
We consider a particular discretization of the harmonic oscillator which admits an orthogonal basis of eigenfunctions called Kravchuk functions possessing appealing properties from the numerical point of view. We analytically prove the…
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of…
We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in…
The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given…
Discrete interaction models for the classical harmonic oscillator are used for introducing new mathematical generalizations in the usual continuous formalism. The inverted harmonic potential and generalized discrete hyperbolic and…
Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits…
The equilibrium properties of an open harmonic oscillator are considered in three steps: First the creation and destruction operators are generalized for open dynamics and the creation operator is used to construct coherent states. The…
Motivated by improving the understanding of the quantum-to-classical transition we use a simple model of classical discrete interactions for studying the discrete-to-continuous transition in the classical harmonic oscillator. A parallel is…
We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that…
A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional $\mathbb{N}\times \mathbb{N}$ lattice. It is governed by a Hamiltonian…
In quantum theory, observables with a continuous spectrum are known to be fundamentally different from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for…
A prepotential approach to constructing the quantum systems with dynamical symmetry is proposed. As applications, we derive generalizations of the hydrogen atom and harmonic oscillator, which can be regarded as the systems with…
Instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discrete set of modes. We derive analytical error bounds on expectation values of system…
The quantum modes of a new family of relativistic oscillators are studied by using the supersymmetry and shape invariance in a version suitable for (1+1) dimensional relativistic systems. In this way one obtains the Rodrigues formulas of…
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…
In two papers we proposed a continuum model for the dynamics of systems of self propelling particles with kinematic constraints on the velocities and discussed some of its properties. The model aims to be analogous to a discrete algorithm…
We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bounded, whereas the spectra of position and momentum are a denumerable non-degenerate set of points in [-1,1] that depends on the…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
We study the connection between Lagrangian and Hamiltonian descriptions of closed/open dynamics, for a collection of particles with quadratic interaction (closed system) and a sub-collection of particles with linear damping (open system).…