A numerical scheme is suggested for accurate large-scale quantum dynamics simulations. The time-dependent Schrödinger equation with finite time-dependent interaction terms is replaced by an inhomogeneous equation with imaginary boundary operators applied along the time axis. This equation is solved globally for a finite time interval using recent Krylov subspace-based iterative methods that are...

We derive exact solutions of the Ablowitz-Ladik (A-L) equation using a special ansatz that linearly relates the real and imaginary parts of the complex function. This ansatz allows us to derive a family of first-order solutions of the A-L equation with two independent parameters. This novel technique shows that every exact solution of the A-L equation has a direct analog among first-order solut...

I present simple analytical methods for computing the properties of ground and excited states of Bose-Einstein condensates, and compare their results to extensive numerical simulations. I consider the effect of vortices in the condensate for both positive and negative scattering lengths, a, and find an analytical expression for the large-N0 limit of the vortex critical frequency for a > 0, by a...

Continuation methods are capable of finding multiform solutions by tracking solution curves. However, these methods may fail to track some desired solution curves due to the interference of the rotational equivalent solutions on a radially symmetric domain. We propose a rotational quotient procedure that applies extra constraints to standard continuation which overcomes this difficulty. We solv...

Multi-symplectic methods have recently been considered as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. The symplectic of Hamiltonian systems is well known, but for Partial Differential Equation (PDEs) this is a global property. In addition, many PDEs can be written as Multisymplectic systems, in which each independent variable has a distinct symplectic structure. ...

The belief propagation algorithm has been recognized in the information theory community as a soft-decision iterative decoding algorithm. It is the most powerful algorithm found so far for attacking hard optimization problems in channel decoding. Quantum mechanics is the foundation of modern physics with the time-independent Schrödinger equation being one of the most important equations. This p...

The classification of short hydrogen bonds depends on several factors including the shape and energy spacing between the nuclear eigenstates of the hydrogen. Here, we describe the NuSol program in which three classes of algorithms were implemented to solve the 1D, 2D and 3D time independent nuclear Schrödinger equation. The Schrödinger equation was solved using the finite differences based Nume...

Effective mass Schrödinger equation is solved exactly for a given potential. NikiforovUvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. On...

The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. We obtain the Schrödinger equation from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton-J...

Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical a...