Related papers: Time behaviour near to spectral singularities
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schr\"odinger equation is an important and challenging problem in quantum mechanics and…
The complete solutions of the Schr\"odinger equation for a particle with time-dependent mass moving in a time-dependent linear potential are presented. One solution is based on the wave function of the plane wave, and the other is with the…
We prove unique continuation properties for solutions of the evolution Schr\"odinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up…
The dynamics of models described by a one-dimensional discrete nonlinear Schr\"odinger equation is studied. The nonlinearity in these models appears due to the coupling of the electronic motion to optical oscillators which are treated in…
We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schr\"{o}dinger equation, we prove that the wave packet…
The Schr\"{o}dinger equation admits smooth and finite solutions that spontaneously evolve into a singularity, even for a free particle. This blowup is generally ascribed to the intrinsic dispersive character of the associated time…
In this paper, we study the decoherence of a wave described by the solution to a Schroedinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze…
Eigenvectors of decaying quantum systems are studied at exceptional points of the Hamiltonian. Special attention is paid to the properties of the system under time reversal symmetry breaking. At the exceptional point the chiral character of…
Results are presented for the time evolution of fermions initially in a non-zero temperature normal phase, following the switch on of an attractive interaction. The dynamics are studied in the disordered phase close to the critical point,…
Open quantum and wave systems exhibit exotic degeneracies at exceptional points in parameter space that have attracted considerable attention in various fields of physics, including optics and photonics. One reason is the strong response of…
A nonlinear scattering transform is studied for the two-dimensional Schrodinger equation at zero energy with a radial potential. First explicit examples are presented, both theoretically and computationally, of potentials with nontrivial…
We present a numerical study of a derivative nonlinear Schr\"odinger equation with a general power nonlinearity, $|\psi|^{2\sigma}\psi_x$. In the $L^2$-supercritical regime, $\sigma>1$, our simulations indicate that there is a finite time…
Asymptotics of solutions to Schroedinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular…
The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case…
In this paper, we investigate properties of unique continuation for hyperbolic Schr\"odinger equations with time-dependent complex-valued electric fields and time-independent real magnetic fields. We show that positive masses inside of a…
The purpose of this document is to describe the solution and implementation of the time-independent and time-dependent Schr\"odinger using pseudospectral methods. Currently, the description is for single particle systems interacting with a…
Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi…
We consider d-dimensional time dependent Schr\"odinger equations on the Hilbert space of square integrable functions. We assume magnetic and scalar potentials are almost critically singular with respect to spatial variables both locally and…
In the context of quantum mechanics superoscillations, or the more general supershifts, appear as initial conditions of the time dependent Schr\"odinger equation. Already in \cite{ABCS21_2} a unified approach was developed, which yields…
We obtain unique continuation results for Schrodinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear…