Related papers: Determining the potential and the gradient couplin…
The static second hyperpolarizability is derived from the space-fractional Schr\"{o}dinger equation in the particle-centric view. The Thomas-Reiche-Kuhn sum rule matrix elements and the three-level ansatz determines the maximum second…
We study an inverse problem related to the dynamical Schr{\"o}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schr\"odinger equation possesses a trivial solution, we linearize the equation around the…
We consider an inverse problem of recovering the unknown coefficients $\beta(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schr\"odinger equation $ (\mathrm{i} \partial_t +\Delta +V)u + \beta u^2=0$ in $(0,T) \times M$, on…
We discuss the extension of the Lewis and Riesenfeld method of solving the time-dependent Schr\"odinger equation to cases where the invariant has continuous eigenvalues and apply it to the case of a generalized time-dependent inverted…
The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. This problem was formulated by Caro and Ruiz and…
We investigate the inverse problem of retrieving the magnetic potential of an Iwatsuka Hamiltonian by knowledge of the second component of the quantum velocity. We show that knowledge of the quantum currents carried by a suitable set of…
The problem of confinement of spinless particles in 1+1 dimensions is approached with a linear potential by considering a mixing of Lorentz vector and scalar couplings. Analytical bound-states solutions are obtained when the scalar coupling…
We prove a local Lipschitz stability estimate for Gel'fand-Calder\'on's inverse problem for the Schr\"odinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the…
The domain of validity of the higher-order Schrodinger equations is analyzed for harmonic-oscillator and Coulomb potentials as typical examples. Then the Cauchy theory for higher-order Hartree-Fock equations with bounded and Coulomb…
We investigate the increasing stability of the inverse Schr\"{o}dinger potential problem with integer power type nonlinearities at a large wavenumber. By considering the first order linearized system with respect to the unknown potential…
We investigate bound state solutions of the 2D Schr\"odinger equation with a dipole potential originating from the elastic effects of a single edge dislocation. The knowledge of these states could be useful for understanding a wide variety…
We discuss a new completely integrable case of the time-dependent Schroedinger equation in $R^n$ with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator…
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and…
In this work, we investigate the shape identification and coefficient determination associated with two time-dependent partial differential equations in two dimensions. We consider the inverse problems of determining a convex polygonal…
The Schr\"odinger-like equation written in terms of the displacement operator is solved analytically for a inverse square plus Coulomb-like potential. Starting from the new Hamiltonian, the effects of the spatially dependent mass on the…
In this paper, we introduce an inverse problem of a Schr\"odinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial…
In this paper we prove stable determination of an inverse boundary value problem associated to a magnetic Schr\"odinger operator assuming that the magnetic and electric potentials are essentially bounded and the magnetic potentials admit a…
The inverse scattering problem for the Schr$\mathrm{\ddot{o}}$dinger operators on the line is considered when the potential is real valued and integrable and has a finite first moment. It is shown that the potential on the line is uniquely…
We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schr\"odinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016)…
We study positive bound states for the semiclassical stationary nonlinear Schr\"odinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows…