Related papers: Unique Continuation for the Magnetic Schr\"odinger…
We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the…
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…
We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive…
We consider all pure or mixed states of a quantum many-body system which exhibit the same, arbitrary but fixed measurement outcome statistics for several commuting observables. Taking those states as initial conditions, which are then…
A fundamental belief is that the formulism of relativistic quantum mechanics equations (RQMEs) should remain in low momentum motion. However, it is found that some formulas from RQMEs were lost in Schr\"odinger equation. For example, a free…
We study equations driven by Schr\"odinger operators consisting of a self-adjoint Dirichlet operator and a singular potential, which belongs to a class of positive Borel measures absolutely continuous with respect to a capacity generated by…
Solution of the Schr\"odinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB$_0$ method. Solution in the whole region is built with the help of simple connection…
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…
This paper investigates the inverse scattering problem for the magnetic Schr\"odinger equation. We first establish the well-posedness of the direct problem through a variational approach under physically meaningful assumptions on the…
We give a sharp upper bound on the vanishing order of solutions to Schrodinger equation with C^1 electric and magnetic potentials on a compact smooth manifold. Our method is based on quantitative Carleman type inequalities developed by…
The purpose of this Comment is to show that the solutions to the zero energy Schr\"odinger equations for monomial central potentials discussed in a recently published Letter, may also be obtained from the corresponding free particle…
The consistency of the concept of quantum (quasi)particles possessing effective mass which is both position- and excitation-dependent is analyzed via simplified models. It is shown that the system may be stable even when the effective mass…
This paper is concerned with the existence and qualitative properties of positive ground state solutions for the planar Schr\"odinger-Newton equation on the disc. First, we prove the existence and radial symmetry of all the positive ground…
We prove unique continuation properties for solutions of evolution Schr\"odinger equation with time dependent potentials. In the case of the free solution these correspond to uncertainly principles referred to as being of Morgan type. As an…
We study the many-body problem of charged particles interacting with their self-generated electromagnetic field. We model the dynamics of the particles by the many-body Maxwell-Schr\"odinger system, where the particles are treated quantum…
We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and…
We consider a system of ODE in a Fr\'echet space with unconditional Schauder basis. The right side of the ODE is a discontinuous function. Under certain monotonicity conditions we prove an existence theorem for the corresponding initial…
Schr\"odinger equation with given, {\it a priori} known current is formulated. A non-zero current density is maintained in the quantum system via a subsidiary condition imposed by vector, local Lagrange multiplier. Constrained minimization…
With a number of special Hamiltonians, solutions of the Schr\"{o}dinger equation may be found by separation of variables in more than one coordinate system. The class of potentials involved includes a number of important examples, including…
We extend our finite difference time domain method for numerical solution of the Schrodinger equation to cases where eigenfunctions are complex-valued. Illustrative numerical results for an electron in two dimensions, subject to a confining…