Related papers: Fixed Energy R-separation for Schroedinger equatio…
Given a potential $V$ and the associated Schr\"odinger operator $-\Delta+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable…
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…
An analogy of the Fokker-Planck equation (FPE) with the Schr\"odinger equation allows us to use quantum mechanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the…
In this study, the Schrodinger equation for flat potentials through the pseudo-perspective method is investigated.
Energy diffusion due to spontaneous localization (SL) for a relativistically-fast moving particle is examined. SL is an alternative to standard quantum theory in which quantum state reduction is treated as a random physical process which is…
We propose a new variational approach to finding multiple critical points for strongly indefinite problems without assuming the weak upper semicontinuity on the variational functionals. By this approach, we obtain the existence of…
We consider stationary $p$-Schr\"odinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and…
Using classical statistics, Schrodinger equation in quantum mechanics is derived from complex space model. Phase-space probability amplitude, that can be defined on classical point of view, has connections to probability amplitude in…
The one-dimensional Schr\"{o}dinger equation for a class of potentials $V(|x|)$ which vanish at infinity and present dominant singularity at the origin in the form $\alpha /|x|^{\beta}$ ($0<\beta \leq 2$) is investigated. The Hermiticity of…
This paper aims to establish the existence of a weak solution for the following problem: \begin{equation*} (-\Delta)^{s}_{\mathcal{H}}u(x) +V(x)h(x,x,|u|)u(x)=\left(\int_{\mathbb{R}^{N}}\dfrac{K(y)F(u(y))}{|x-y|^\lambda}dy \right)…
We study the energy and momentum of an isolated system in the tetrad theory of gravitation, starting from the most general Lagrangian quadratic in torsion, which involves four unknown parameters. When applied to the static spherically…
This article concerns the fractional elliptic equations \begin{equation*}(-\Delta)^{s}u+\lambda V(x)u=f(u), \quad u\in H^{s}(\mathbb{R}^N), \end{equation*}where $(-\Delta)^{s}$ ($s\in (0\,,\,1)$) denotes the fractional Laplacian, $\lambda…
We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group. Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is…
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…
The interrelationship between energy and probability conservation is explored from the point of view of statistical physics and non-relativistic quantum mechanics. The simultaneous validity of the law of conservation of energy and the…
The Parisi formula for the free energy of the Sherrington-Kirkpatrick model is completed to a closed-form generating functional. We first find an integral representation for a solution of the Parisi differential equation and represent the…
We consider the relativistic Schr\"odinger equation with a time dependent vector and scalar potential on a bounded cylindrical domain. Using a Geometric Optics Ansatz we establish a logarithmic stability estimate for the recovery of the…
We here show how the methods recently applied by [DW16] to solve the stochastic nonlinear Schr\"odinger equation on $\mathbb{T}^2$ can be enhanced to yield solutions on $\mathbb{R}^2$ if the non-linearity is weak enough. We prove that the…
The formulation of the eigenvalue problem for the Schr\"odinger equation is studied, for the numerical solution a new approach is applied. With the usual exponentially rising free-state asymptotical behavior, and also with a first order…
The Schr\"odinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into…