Related papers: Exact boundary conditions at finite distance for t…
We review an explicit approach to obtaining numerical solutions of the Schr\"odinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with…
This study presents a numerical simulation of a quantum electron confined in a 10 nm potential well, using the Crank-Nicolson numerical technique to solve the time-dependent Schrodinger equation. The results capture the evolution of the…
In this article, we present the mathematical analysis of the convergence of the linearized Crank-Nicolson Galerkin method for a nonlinear Schrodinger problem related to a domain with a moving boundary. The convergence analysis of the…
In this paper we study the numerical method and the convergence for solving the time-dependent Maxwell-Schr\"{o}dinger equations under the Lorentz gauge. An alternating Crank-Nicolson finite element method for solving the problem is…
We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schr\"odinger equation. On one hand, we show that a closed quantum system is defined by local boundary…
A new energy-based stochastic extension of the Schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closed-form solution to the dynamical equation, valid for all…
We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schr\"odinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control…
Exploring the idea that equation for radial wave function must be compatible with the full Schrodinger equation, a boundary condition is derived.
It is well-known that time-dependent Schr\"{o}dinger equation can only be exactly solvable in very rare cases, even for two-level quantum systems. Therefore, finding exact quantum dynamics under time-dependent Hamiltonian is not only of…
In this paper, we study the schrodinger equation and wave equation with the Dirichlet boundary condition on a connected finite graph. The explicit expressions for solutions are given and the energy conservations are derived. Applications to…
We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals of Crank-Nicolson type. The feature of this functional is that it enjoys energy conservation in the absence of free boundaries, which is an…
The time-dependent Schrodinger equation is solved for two model problems for a non-Hermitian quantum system.A simple matrix model system is used to examine two critical problems for these systems: complex and non-observable energies and…
A closed expression for the harmonic oscillator wave function after the passage of a linear signal with arbitrary time dependence is derived. Transition probabilities are simple to express in terms of Laguerre polynomials. Spontaneous…
In this paper, a new analysis for existence, uniqueness, and regularity of solutions to a time-dependent Kohn-Sham equation is presented. The Kohn-Sham equation is a nonlinear integral Schroedinger equation that is of great importance in…
We study the nonlinear Schr\"odinger equation on the half-line with a boundary condition that involves time derivative. This boundary condition was presented by Zambon [J. High Energ. Phys. 2014 (2014) 36]. We establish the integrability of…
This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential…
We obtain exact solutions to the class of parabolic partial differential equations of arbitrary dimensionality and with arbitrary potentials. The solutions are presented in a compact-form: as explicit mathematical expressions consisting of…
We prove necessary optimality conditions for problems of the calculus of variations on time scales with a Lagrangian depending on the free end-point.
In this paper, we study loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference scheme of Crank-Nicholson type of higher order is developed for…
In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell--Schr\"{o}dinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an…