Related papers: Extended Weyl Calculus and Application to the Phas…
New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of…
We derive the functional Schrodinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a…
A periodic two-phase algebro-geometric solution of the focusing nonlinear Schr\"odinger equation is constructed in terms of elliptic Jacobi theta-functions. A dependence of this solution on the parameters of a spectral curve is…
We carry out the canonical quantization of the Levi-Civit\`a family of static and cylindrical solutions. The reduced phase space of this family of metrics is proved to coincide with that corresponding to the Kasner model, including the…
The phase space of quantum mechanics can be viewed as the complex projective space endowed with a Kaehlerian structure given by the Fubini-Study metric and an associated symplectic form. We can then interpret the Schrodinger equation as…
We develop a phase-space framework for fractional generalised anharmonic oscillators and their heat semigroups on weighted modulation spaces. We consider operators of the form \[ \mathcal{H}_{k,l}=(-\Delta)^{l}+V(x), \] where $V$ is a…
We prove the following extension of the Wiener--Wintner Theorem in Ergodic Theor and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows $ (X,\mu , T_t)$ and $ f\in L^p (X,\mu)$, there is a set…
The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard structure on the Euclidean phase space. In this paper we describe the corresponding algebra of…
In this paper we bring into attention variable coefficient cubic-quintic nonlinear Schr\"odinger equations which admit Lie symmetry algebras of dimension four. Within this family, we obtain the reductions of canonical equations of…
We consider the one-dimensional Schr\"odinger equation $-f''+q_\alpha f = Ef$ on the positive half-axis with the potential $q_\alpha(r)=(\alpha-1/4)r^{-2}$. It is known that the value $\alpha=0$ plays a special role in this problem: all…
We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schroedinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator)…
We consider the pointwise convergence problem for the solution of Schr\"odinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson's problem as the most simple case and was…
The numerical simulation of the time-dependent Schr\"odinger equation for quantum systems is a very active research topic. Yet, resolving the solution sufficiently in space and time is challenging and mandates the use of modern…
The so-called Weyl transform is a linear map from a commutative algebra of functions to a noncommutative algebra of linear operators, characterized by an action on Cartesian coordinate functions of the form $(x, y) \mapsto (X, Y)$ such that…
Canonical coordinates for the Schr\"odinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schr\"odinger equation, considered as a classical field theory, shares with Liouville completely…
This paper investigates the functional calculus of the harmonic oscillator on each Moyal-Groenewold plane, the noncommutative phase space which is a fundamental object in quantum mechanics. Specifically, we show that the harmonic oscillator…
In this paper we have generalized the quantum mechanics on fractal time-space. The time is changing on Cantor-set like but space is considered as fractal curve like Von-Koch curve. The Feynman path method in quantum mechanics has been…
A Schr\"odinger type equation for a mathematical probability amplitude {\psi}(x,t), is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M, moving in a potential V(x). The…
We derive an explicit formula for the general solution of the cubic Szeg\"o equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel…
Maxwell's equations in the vacuum can be formally cast in the form of Schr\"odinger's equation. Unfortunately, the vector to which this equation directly applies is not a wavefunction: its amplitude squared is not a probability density but…