Related papers: On Time-Dependant Symmetries of Schroedinger Equat…
We consider the time-dependent non linear Schrodinger equations with a double well potential in dimensions d =1 and d=2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues…
We study arbitrary order symmetry operators for the linear Schr\"odinger equations with arbitrary number of spatial variables. We deduce determining equations for coefficient functions of such operators and consider in detail some cases…
We prove that the spectrum of an n-dimensional semiclassical radial Schr\"odinger operator determines the potential within a large class of potentials for which we assume no symmetry or analyticity. Our proof is based on the first two…
We show that the measure of the spectrum of Schr\"odinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a…
We consider time-dependent Schroedinger equations in one dimension with double well potential and an external nonlinear perturbation. If the initial state belongs to the eigenspace spanned by the eigenvectors associated to the two lowest…
We study symmetry properties of the Schr\"odinger equation with the potential as a new dependent variable, i.e., the transformations which do not change the form of the class of equations. We also consider systems of the Schr\"odinger…
Some symmetries of time-dependent Schr\"odinger equations for inverse quadratic, linear, and quadratic potentials have been systematically examined by using a method suitable to the problem. Especially, the symmetry group for the case of…
We show that for a Schr\"odinger operator with bounded potential on a manifold with cylindrical ends the space of solutions which grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or,…
We consider Schr\"odinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials…
We prove necessary and sufficient conditions for the Schr\"odinger operators to have zero-energy bound states at the threshold of the essential spectrum such that they have bounded $k$-th moment. This result is the extension of the results…
In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schr\"odinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away…
We discuss the relationship between exact solvability of the Schroedinger equation, due to a spatially dependent mass, and the ordering ambiguity. Some examples show that, even in this case, one can find exact solutions. Furthermore, it is…
We show that whole-line Schr\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential…
In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schr\"odinger equation for systems with potentials $V(x,\tau)=g^{(2)}(\tau)x^2+g^{(1)}(\tau)x +g^{(0)}(\tau)$. We…
We study singular Schr\"odinger operators on a finite interval as selfadjoint extensions of a symmetric operator. We give sufficient conditions for the symmetric operator to be in the $n$-entire class, which was defined in our previous…
We show that there exist pairs of non-isometric potentials for the 1D semiclassical Schr\"odinger operator whose spectra agree up to $O(h^\infty)$, yet their corresponding eigenvalues differ no less than exponentially. This result was…
We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…
Two systems for a charged particle are studied, the first one when it is under the effect of a constant electric field, and the second one when it is under the effect of a constant electromagnetic field. For both systems, it is possible to…
Within the framework of self-adjoint operator of time in non-relativistic quantum mechanics some properties of solutions of Schroedinger equation, related to Hilbert space formalism, are investigated for two types of time dependent…
The Schroedinger operator on the Newtonian space-time is defined in a way which is independent on the class of inertial observers. In this picture the Schroedinger operator acts not on functions on the space-time but on sections of certain…