Related papers: Higher order Schrodinger and Hartree-Fock equation…
In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schr\"odinger operators. We deduce the strong unique continuation property in the presence of subcritical and…
In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of Schr\"{o}dinger equations with Hardy potentials: \begin{equation*} \left\{ \begin{aligned}…
In this paper we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result…
Following Symanzik we argue that the Schr\"odinger functional in lattice gauge theories without matter fields has a well-defined continuum limit. Due to gauge invariance no extra counter terms are required. The Schr\"odinger functional is,…
We study the quantum mechanics of the derivative nonlinear Schrodinger equation which has appeared in many areas of physics and is known to be classically integrable. We find that the N-body quantum problem is exactly solvable with both…
Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a second…
Quantum scattering by a one-dimensional odd potential proportional to the square of the distance to the origin is considered. The Schr\"odinger equation is solved exactly and explicit algebraic expressions of the wavefunction are given. A…
We consider the homogenization at second-order in $\varepsilon$ of $\mathbb{L}$-periodic Schr\"odinger operators with rapidly oscillating potentials of the form $H^\varepsilon =-\Delta + \varepsilon^{-1} v(x,\varepsilon^{-1}x ) + W(x)$ on…
An analysis is given of the local phase space of gravity coupled to matter to second order in perturbation theory. Working in local regions with boundaries at finite distance, we identify matter, Coulomb, and additional boundary modes. The…
The Schrodinger equation for stationary states in a central potential is studied in an arbitrary number of spatial dimensions, say q. After transformation into an equivalent equation, where the coefficient of the first derivative vanishes,…
We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing…
Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential…
Recently developed simple approach for the exact/approximate solution of Schrodinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one…
We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate…
We present a new six-parameter family of potentials whose solutions are expressed in terms of the hypergeometric functions 3F2, 2F2 and 1F2. Both the scattering data and the bound states of these potentials are explicitly computed and the…
We prove that the Schr\"odinger equation for N number of particles in the time dependent electro-magnetic field generates a unique unitary propagator on the state space under the condition that the field is smooth and moderately but almost…
Similarly to the interaction lagrangian, the possible boundary conditions in quantum field theories on space-time manifolds with boundaries are strongly constrained by the symmetry and scaling properties of the theory. Based on this general…
Quantum lattice models describe a wide array of physical systems, and are a canonical way to numerically solve the Schrodinger equation. Here we prove the potential inversion theorem, which says that wavefunction probability in these models…
We consider magnetic Schr\"odinger equations with sublinear magnetic potentials and subquadratic electric potentials on $\mathbb{R}^{d}$, as well as generalizations thereof. We obtain new results on the global well-posedness of the Cauchy…
In this paper, we consider solutions to the following nonlinear Schr\"odinger equation with competing Hartree-type nonlinearities, $$ -\Delta u + \lambda u=\left(|x|^{-\gamma_1} \ast |u|^2\right) u - \left(|x|^{-\gamma_2} \ast |u|^2\right)…