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Using the simplest but fundamental example, the problem of the infinite potential well, this paper makes an ideological attempt (supported by rigorous mathematical proofs) to approach the issue of…
In this article, we propose a modified nonlinear Schr\"odinger equation for modeling pulse propagation in optical waveguides. The proposed model bifurcates into a system of elliptic and hyperbolic equations depending on waveguide…
Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schr\"odinger equation. In this paper we present three strategies for extending any sixth-order scheme for…
In this paper, we numerically address the inverse problem of identifying a time-dependent coefficient in the time-fractional diffusion equation. An a priori estimate is established to ensure uniqueness and stability of the solution. A fully…
A new hybridizable discontinuous Galerkin method, named the CHDG method, is proposed for solving time-harmonic scalar wave propagation problems. This method relies on a standard discontinuous Galerkin scheme with upwind numerical fluxes and…
The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to…
Using a hybrid approach, based on the recursion relations for shape invariant potentials developed by Das and Huang and a time-dependent tranformation of variables, we derive the propagator for a radial oscillator. Although this is not a…
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…
Linear combinations of complex gaussian functions, where the linear and nonlinear parameters are allowed to vary, are shown to provide an extremely flexible and effective approach for solving the time-dependent Schr\"odinger equation in one…
How to accurately solve time-dependent Schr\"odinger equation is an interesting and important problem. Here, we propose a novel method to obtain the exact Floquet solutions of the Schr\"odinger equation for periodically driven systems by…
We consider the time-dependent Schr\"odinger equation on a Riemannian manifold $\mathcal{A}$ with a potential that localizes a certain class of states close to a fixed submanifold $\mathcal{C}$. When we scale the potential in the directions…
A variationally improved Sturmian approximation for solving time-independent Schr\"odinger equation is developed. This approximation is used to obtain the energy levels of a quartic anharmonic oscillator, a quartic potential, and a Gaussian…
In this work we present the theoretical framework for the solution of the time-dependent Schr\"{o}dinger equation (TDSE) of atomic and molecular systems under strong electromagnetic fields with the configuration space of the electron's…
The Schwinger-Dyson equation for a scalar propagator is solved in Minkowski space with the help of an integral spectral representation, both for spacelike and timelike momenta. The equation is re-written into a form suitable for numerical…
The linear Schr\"odinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with…
We prove a dispersive estimate for periodic discrete Schr\"odinger operators on the line with optimal rate of decay. Additionally, by standard methods, we deduce dispersive estimates for the discrete nonlinear Schr\"odinger equation with…
We construct explicit integral relations between propagators of generalized Schr\"odinger equations that are linked by higher-order supersymmetry. Our results complement and extend the findings obtained in J. Phys.A 40, 10557 (2007) for the…
We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schr\"odinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time…
A new type of solution for the full 3+1 dimensional space-time Schroedinger equation is presented here. We consider elegant presentation of the exact solution in a spherical coordinate system, along with the assuming of separation of the…
We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schroedinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion…