Related papers: Singular inverse square potential in coordinate sp…
We study the observability properties of the Grushin equation with an inverse square potential, whose singularity occurs at the boundary of two-dimensional rectangular domains or in the interior of the domain in higher dimensions. In some…
We prove that an $L^\infty$ potential in the Schr\"odinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace…
In this work we study the Schr\"{o}dinger equation in the presence of the Hartmann potential with a generalized uncertainty principle. We pertubatively obtain the matrix elements of the hamiltonian at first order in the parameter of…
We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential, after the Eckart and P\"oschl-Teller potentials, which is proportional to an arbitrary variable parameter and has a shape that…
I investigate spacetime singularities from the point of view of the wavefunction of the universe. In order to extend the classical notion of geodesic incompleteness one has to include the proper time of an observer as a degree of freedom in…
We present the exact solution of the stationary Schr\"odinger equation equation for the potential $V=V_0/{\sqrt{x}}$. Each of the two fundamental solutions that compose the general solution of the problem is given by a combination with…
In this paper, we study the scattering theory for the cubic inhomogeneous Schr\"odinger equations with inverse square potential $iu_t+\Delta u-\frac{a}{|x|^2}u=\lambda |x|^{-b}|u|^2u$ with $a>-\frac14$ and $0<b<1$ in dimension three. In the…
We give a lower bound for the energy of a quantum particle in the infinite square well. We show that the bound is exact and identify the well-known element that fulfils the equality. Our approach is not directly dependent on the…
We study the inverse backscattering problem for the Schr\"odinger equation in two dimensions. We prove that, for a non-smooth potential in 2D the main singularities up to 1/2 of the derivative of the potential are contained in the Born…
Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, we obtain a large class of new shape invariant potentials which are…
We consider Calder\'{o}n's inverse boundary value problems for a class of nonlinear Helmholtz Schr\"{o}dinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the…
We provide necessary and sufficient geometric conditions for the exact observability of the Schr\"odinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for…
Starting from the semi-classical spectrum of Schr\"odinger operators $-h^2\Delta+V$ (on $\mathbb{R}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express…
A one-dimensional generalized nonlinear Schroedinger equation is considered, and the corresponding inverse scattering problem is analyzed when the potential is compactly supported and depends on the wave function. The unique recovery of the…
We consider the inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;\mathcal{L}_a=\Delta -\frac{a}{|x|^2}, $$ where $\lambda=\pm1$,…
We provide a characterization of the spectral minimum for a random Schr\"odinger operator of the form $H=-\Delta + \sum_{i \in \Z^d}q(x-i-\omega_i)$ in $L^2(\R^d)$, where the single site potential $q$ is reflection symmetric, compactly…
We consider phaseless inverse scattering for the Schr\"odinger equation with compactly supported potential in dimension $d\ge 2$. We give explicit formulas for solving this problem from appropriate data at high energies. As a corollary, we…
The Heun's equation with its four regular singularities emerges in many applications in science. Despite the growing interest of the scientific community, the literature has many gaps in conceptual mathematical aspects of this equation.…
Starting from a continuum theory of defects, that is the analogous to three-dimensional Einstein-Cartan-Sciama-Kibble gravity, we consider a charged particle with spin 1/2 propagating in a uniform magnetic field coincident with a wedge…
We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schr\"odinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of…