Related papers: Riccati equations and quasi-1D noninteracting prob…
We construct the Green functions (or Feynman's propagators) for the Schroedinger equations of the form $i\psi_{t}+{1/4}\psi_{xx}\pm tx^{2}\psi =0$ in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and…
A new parametrisation of the Eilenberger equations of superconductivity in terms of the solutions to a scalar differential equation of the Riccati type is introduced. It is shown that the quasiclassical propagator, and in particular the…
We study the one-dimensional Schr\"odinger equation and derive exact expressions for the Green function in terms of reflection coefficients which are defined for semi-infinite intervals. We also discuss the relation between our results and…
Calculation of current and order parameter distribution in inhomogeneous superconductors is often based on a self-consistent solution of Eilenberger equations for quasiclassical Green's functions. Compared to the original Gorkov equations,…
A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie…
In this work we relate the spectral problem of the toroidal six-vertex model's transfer matrix with the theory of integrable non-linear differential equations. More precisely, we establish an analogy between the Classical Inverse Scattering…
The study of the Schr\"{o}dinger equation with the position-dependent effective mass has attracted a lot of attention, due to its applications in many fields of physics, including the properties of the semiconductors, semiconductor…
The central role played by pseudodifferential operators in relativistic dynamics is very well know. In this work, operators as the Schrodinger one (e.g: square root) are treated from the point of view of the non-local pseudodifferential…
We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati…
Here we review the many aspects and distinct phenomena associated to quantum dynamics on general graph structures. For so, we discuss such class of systems under the energy domain Green's function ($G$) framework. This approach is…
We consider the variational problem of maximizing the weighted equilibrium Green's energy of a distribution of charges free to move in a subset of the upper half-plane, under a particular external field. We show that this problem admits a…
High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schr\"{o}dinger equation is cast into nonlinear Riccati equation, which is solved analytically in first…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…
We study the full set of planar Green's functions for a two-matrix model using the language of functions of non-commuting variables. Both the standard Schwinger-Dyson equations and equations determining connected Green's functions can be…
A classical problem of free-space Green's function $G_{0\Lambda}$ representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension…
Time-dependent quantum mechanics provides an intuitive picture of particle propagation in external fields. Semiclassical methods link the classical trajectories of particles with their quantum mechanical propagation. Many analytical results…
In this work, we generalize previous results about the Fractionary Schr\"{o}dinger Equation within the formalism of the theory of Tempered Ultradistributions. Several examples of the use of this theory are given. In particular we evaluate…
A covariant non-local extention if the stationary Schr\"odinger equation is presented and it's solution in terms of Heisenbergs's matrix quantum mechanics is proposed. For the special case of the Riesz fractional derivative, the calculation…
We consider Schr\"odinger equations and Fokker-Planck equations in one dimension, and study the low-energy asymptotic behavior of the Green function using a new method. In this method, the coefficient of the expansion in powers of the wave…
We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schr\"odinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation,…