In this work, we follow the idea of the De Broglie’s matter waves and the analogous method that Schrödinger founded wave equation, but we apply the more essential Hamilton principle instead of the minimum action principle of Jacobi which was used in setting up Schrödinger wave equation. Thus, we obtain a novel non-relativistic wave equation which is different from the Schrödinger equation, and ...

We show that for a one-dimensional Schrödinger operator with a potential, whose (j + 1)-th moment is integrable, the j-th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrödinger equation in the resonant case.

The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schrödinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton’s equations also give the Schrödinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order de...

interaction and correlation effects in quantum dots play a fundamental role in defining both their equilibrium and transport properties. numerical methods are commonly employed to study such systems. in this paper we investigate the numerical calculation of quantum transport of electrons in spherical centered defect ingaas/algaas quantum dot (scdqd). the simulation is based on the imaginary time...

In a recent paper [1] it was shown a clean connection between the solution of the classical heat equation and the Tartaglia/Pascal/Yang-Hui triangle. When the time variable in the heat equation is substituted with the imaginary time, the heat equation becomes the Schrödinger equation of the quantum mechanics. So, a conjecture was put forward about a connection between the solution of the Schröd...

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n ≥ 6.

We prove unique continuation properties for solutions of the evolution Schrödinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi-linear Schrödinger equations.

The time-dependent Schrödinger equation for two quite general types of perturbation has been solved by introducing the Laplace transforms to eliminate the time variable. The resulting time-independent differential equation can then be solved by the perturbation method, the variation method, the variation-perturbation method, and other methods.

Since the kinetic and potential energy terms of the real-time nonlinear Schrödinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well-known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even ...

Quantum stochastic methods based on effective wave functions form a framework for investigating the generally non-Markovian dynamics of a quantum-mechanical system coupled to a bath. They promise to be computationally superior to the master-equation approach, which is numerically expensive for large dimensions of the Hilbert space. Here, we numerically investigate the suitability of a known sto...