The interaction of intense ultrashort laser pulses with a gas of argon atoms is numerically investigated. The investigation utilizes a model that solves both the time-dependent Schrödinger equation and the wave equation self-consistently. The time-dependent Schrödinger equation for single-active electron in the effective static potential and in the potential of electron-laser interaction is sol...

A generalized inverse scattering method has been developed for arbitrary n dimensional Lax equations. Subsequently, the method has been used to obtain N soliton solutions of a vector higher order nonlinear Schrödinger equation, proposed by us. It has been shown that under suitable reduction, vector higher order nonlinear Schrödinger equation reduces to higher order nonlinear Schrödinger equatio...

The Schrödinger equation with the central potential is first studied in the arbitrary dimensional spaces and obtained an analogy of the two-dimensional Schrödinger equation for the radial wave function through a simple transformation. As an example, applying an ansatz to the eigenfunctions, we then arrive at an exact closed form solution to the modified two-dimensional Schrödinger equation with...

We analyze the extension of the well known relation between Brownian motion and Schrödinger equation to the family of Lévy processes. We propose a Lévy– Schrödinger equation where the usual kinetic energy operator – the Laplacian – is generalized by means of a pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy–Khintchin formula sh...

Conservation properties of a full discretization via a spectral semi-discretization in space and a Lie-Trotter splitting in time for cubic Schrödinger equations with small initial data (or small nonlinearity) are studied. The approximate conservation of the actions of the linear Schrödinger equation, energy, and momentum over long times is shown using modulated Fourier expansions. The results a...

5.1 The Schrödinger and Heisenberg pictures 5.2 Interaction Picture 5.2.1 Dyson Time-ordering operator 5.2.2 Some useful approximate formulas 5.3 Spin1 precession 2 5.4 Examples: Resonance of a Two-Level System 5.4.1 Dressed states and AC Stark shift 5.5 The wave-function 5.5.1 Position representation 5.5.2 Momentum representation 5.5.3 Schrödinger equation for the wavefunction 5.6 Feynman’s pa...

It is shown that the eigenvalue problem for the Hamiltonians of the standard form, H = p/(2m) + V (x), is equivalent to the classical dynamical equation for certain harmonic oscillators with time-dependent frequency. This is another indication of the central role played by time-dependent harmonic oscillators in quantum mechanics. The utility of the known results for eigenvalue problem in the so...

We prove unique continuation properties for solutions of evolution Schrödinger equation with time dependent potentials. In the case of the free solution these correspond to uncertainly principles referred to as being of Morgan type. As an application of our method we also obtain results concerning the possible concentration profiles of solutions of semi-linear Schrödinger equations.

We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V (x) = ±|x|.

In quantum mechanics, the space-fractional Schrödinger equation provides a natural extension of the standard Schrödinger equation when the Brownian trajectories in Feynman path integrals are replaced by Levy flights. Here an optical realization of the fractional Schrödinger equation, based on transverse light dynamics in aspherical optical cavities, is proposed. As an example, a laser implement...