Related papers: Integrable conditions for Dirac Equation and Schr\…
The Schr\" odinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring…
In this work, we study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+\eta(t)\sum_{j=1}^J\delta_{x=a_j(t)}\psi$ on $L^2((0,1),C)$ where $\eta:[0,T]\longrightarrow R^+$ and $a_j:[0,T]\longrightarrow (0,1)$, $j=1,...,J$. We show how…
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family…
We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schr\"odinger (DDNLS)…
In the framework of the application of the Boundary Control method to solving the inverse dynamical problems for the one-dimensional Schr\"odinger and Dirac operators on the half-line and semi-infinite discrete Schr\"odinger operator, we…
Functional Schr\"{o}dinger equations for interacting fields are solved via rigorous non-perturbative Feynman type integrals.
Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed…
It is noted that the Schrodinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Higher order…
The matrix Schr\"odinger equation is considered on the half line with the general selfadjoint boundary condition at the origin described by two boundary matrices satisfying certain appropriate conditions. It is assumed that the matrix…
In this work, we analytically study the Schr\"odinger equation for the (non-pure) dipolar ion potential V (r) = q/r + Dcos{\theta}/r 2 , in the case of 2D systems using the separation of variables and the Mathieu equations for the angular…
A nonlinear Schr\"odinger equation with external potential $-(t+b)^{-1}$ is considered and its explicit solutions are constructed.
The Dirac delta function potential is considered within the real Hilbert space approach for complex wave functions, as well as quaternionic wave functions. As has been previously determined, the real Hilbert space approach enables the…
The study of integrability of the mathematical physics equations showed that the differential equations describing real processes are not integrable without additional conditions. This follows from the functional relation that is derived…
We consider the relation between the discrete coupled nonlinear Schr\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\"{o}dinger equation. In the same way…
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 construct a variety of new exactly-solvable quantum systems, the potentials of which are given in terms of Lambert-W functions. In particular, we generate Schr\"odinger models with energy-dependent potentials, conventional Schr\"odinger…
We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the…
The Dirac oscillator coupled to an external two-component field can retain its solvability, if couplings are appropriately chosen. This provides a new class of integrable systems. A simplified way of solution is given, by recasting the…
By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schr\"odinger operators has no point…
Approximate analytical solutions of the Dirac equation are obtained for some diatomic molecular potentials plus a tensor interaction with spin and pseudospin symmetries with any angular momentum. We find the energy eigenvalue equations in…