Related papers: On linear Schr\"odinger parabolic problems in Morr…

This paper deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. When the poles form a symmetric structure, it is natural we wonder…

In this paper we prove the existence of meromorphic solutions to a nonlinear differential difference equation that describe certain self-similar potentials for the Schroedinger operator.

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space $ L^{p,\varphi} $ or a Stummel class $ \tilde{S}_{\alpha,p} $. Our results generalize and extend Fefferman's inequalities…

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical…

For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…

We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrodinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of…

For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class…

We study the Cauchy problem for Schr\"odinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in L_{p}-spaces of functions whose regularity is defined by a scalable,…

We obtain exact solutions of the 2D Schr\"odinger equation with the Singular Even-Power and Inverse-Power Potentials in non-commutative complex space, using the Power-series expansion method. Hence we can say that the Schr\"odinger equation…

Using the algebraic approach Lie symmetries of time dependent Schroedinger equations for charged particles interacting with superpositions of scalar and vector potentials are classified. Namely, all the inequivalent equations admitting…

In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schr\"odinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.

Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq3$, where $\Delta$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_s$ for $s\geq d/2$. For…

The Cauchy problem for the Schr\"odinger equations is studied with time-dependent potentials growing polynomially in the spatial direction. First the existence and the uniqueness of solutions are shown in the weighted Sobolev spaces. In…

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…

A non-homogeneous conormal derivative problem is considered for quasilinear divergence form elliptic equations modeled on the $m$-Laplacian operator. The nonlinear terms are given by Carath\'eodory functions and satisfy controlled growth…

This report discusses two new ideas for using perturbation methods to solve the time-independent Schr\"odinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix…

We consider the inverse coefficient problem of simultaneously determining the space dependent electromagnetic potential, the zero-th order coupling term and the first order coupling vector of a two-state Schr\"odinger equation in a bounded…

We obtain exact solutions of the 2D Schr\"odinger equation for Hydrogen atom with the lenear and Harmonic Potentials in noncommutative complex space, using the Power-series expansion method. Hence we can say that the Schr\"odinger equation…