A quenched study of the Schrödinger functional with chirally rotated boundary conditions: Applications

In a previous paper (González López, et al., 2013) [1], we have discussed the non-perturbative tuning of the chirally rotated Schrödinger functional (χSF$\chi \text{SF}$). This tuning is required to eliminate bulk O(a) cutoff effects in physical correlation functions. Using our tuning results obtained in González López et al. (2013) [1] we perform scaling and universality tests analyzing the residual O(a  ) cutoff effects of several step-scaling functions and we compute renormalization factors at the matching scale. As an example of possible application of the χSF$\chi \text{SF}$ we compute the renormalized strange quark mass using large volume data obtained from Wilson twisted mass fermions at maximal twist.     

A class of exactly solvable Schrödinger equation with moving boundary condition

Using first and second order supersymmetry formalism we obtain a class of exactly solvable potentials subject to moving boundary condition.     

Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method

This paper presents the coupled version of a previous work on nonlinear Schrödinger equation [23]. It focuses on the construction of approximate solutions of nonlinear Schrödinger equations. In this paper, we applied the differential transformation method (DTM) to solving coupled Schrödinger equations. The obtained results show that the technique suggested here is accurate and easy to apply.     

Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation

Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formulation, and its connection with the functional Schrödinger representation in the temporal gauge are discussed. The mass gap problem is related to the finite-dimensional spectral problem for a generalized Clifford-valued magnetic Schrödinger operator which represents the DW Hamiltonian operator.     

Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications

In this paper we study blow-up phenomena in general coupled nonlinear Schrödinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several applications of our results to heteronuclear multispecies Bose-Einstein condensates and to degenerate boson-fermion mixtures.     

Asymptotic symmetry of geometries with Schrödinger isometry

We show that the asymptotic symmetry algebra of geometries with Schrödinger isometry in any dimension is an infinite-dimensional algebra containing one copy of Virasoro algebra. It is compatible with the fact that the corresponding geometries are dual to non-relativistic CFTs whose symmetry algebra is the Schrödinger algebra which admits an extension to an infinite-dimensional symmetry algebra containing a Virasoro subalgebra.     

The Bethe ansatz for the six-vertex model with rotated boundary conditions

A method is suggested for derivation of the Bethe ansatz equations for the six-vertex model on a square lattice rotated at an arbitrary angle with respect to the coordinate axes. The method is based on the random walk representation for configurations of the model. The equations for the ice model on the rotated lattice are derived and some numerical results are obtained.     

A General Approach of Quasi-Exactly Solvable Schrödinger Equations

We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schrödinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter our formalism but we focus on quasi-exact interactions for which no such general approach has been considered before. In particular we concentrate on a generalized sextic oscillator but also on the Lamé and the screened Coulomb potentials.     

Analytic treatment of linear and nonlinear Schrödinger equations: A study with homotopy-perturbation and Adomian decomposition methods

In this work, two powerful analytical methods, called homotopy-perturbation method (HPM) and Adomian decomposition method (ADM) are introduced to obtain the exact solutions of linear and nonlinear Schrödinger equations. The main objective is to propose alternative methods of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The results show that these methods are very efficient and convenient and can be applied to a large class of problems. The comparison of the methods shows that although the numerical results of these methods are the same, HPM is much easier, more convenient and efficient than ADM.     

On Hamiltonian Formulations of the Schrödinger System

We review and compare different variational formulations for the Schrödinger field. Some of them rely on the addition of a conveniently chosen total time derivative to the hermitic Lagrangian. Alternatively, the Dirac-Bergmann algorithm yields the Schrödinger equation first as a consistency condition in the full phase space, second as canonical equation in the reduced phase space. The two methods lead to the same (reduced) Hamiltonian. As a third possibility, the Faddeev-Jackiw method is shown to be a shortcut of the Dirac method. By implementing the quantization scheme for systems with second class constraints, inconsistencies of previous treatments are eliminated.     

Rational solutions of the general nonlinear Schrödinger equation with derivative

The double Wronskian solutions whose entries satisfy matrix equation of the general nonlinear Schrödinger equation with derivative (GDNLSE) are derived through the Wronskian technique. Soliton solutions and rational solutions of GDNLSE are obtained by taking special cases in general solutions.     

A parallel algorithm for solving the 3d Schrödinger equation

We describe a parallel algorithm for solving the time-independent 3d Schrödinger equation using the finite difference time domain (FDTD) method. We introduce an optimized parallelization scheme that reduces communication overhead between computational nodes. We demonstrate that the compute time, t, scales inversely with the number of computational nodes as t ∝ (N_nodes)^{−0.95 ± 0.04}. This makes it possible to solve the 3d Schrödinger equation on extremely large spatial lattices using a small computing cluster. In addition, we present a new method for precisely determining the energy eigenvalues and wavefunctions of quantum states based on a symmetry constraint on the FDTD initial condition. Finally, we discuss the usage of multi-resolution techniques in order to speed up convergence on extremely large lattices.     

Dynamics of cubic–quintic nonlinear Schr?dinger equation with different parameters

We study the dynamics of the cubic–quintic nonlinear Schr?dinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions from the quintic term are discussed. We observe the elliptic orbit, homoclinic orbit crossing, quasirecurrence, and stochastic motion with different nonlinear parameters in this system. Numerical simulations show that the changing processes of the motion of the system and the trajectories in the phase space are various for different cubic nonlinear parameters with the increase of the quintic nonlinear parameter.     

Quasi-stationary states of the NRT nonlinear Schrödinger equation

With regards to the nonlinear Schrödinger equation recently advanced by Nobre, Rego-Monteiro, and Tsallis (NRT), based on Tsallis q$q$-thermo-statistical formalism, we investigate the existence and properties of its quasi-stationary solutions, which have the time and space dependences “separated” in a q$q$-deformed fashion. One recovers the normal factorization into purely spatial and purely temporal factors, corresponding to the standard, linear Schrödinger equation, when the deformation vanishes (q=1)$(q=1)$. We discuss various specific examples of exact, quasi-stationary solutions of the NRT equation. In particular, we obtain a quasi-stationary solution for the Moshinsky model, providing the first example of an exact solution of the NRT equation for a system of interacting particles.