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The effects of quantum fluctuations in fields confined by background configurations may be simply and transparently computed using the Green's function approach pioneered by Schwinger. Not only can total energies and surface forces be…
We study the decay of general initial states out of a metastable potential well in quantum mechanics. We provide a closed-form expression for the probability current that tunnels through the barrier in terms of the resonant states into…
We use the factorization method to find the exact eigenvalues and eigenfunctions for a particle in a box with the delta function potential $V(x)=\lambda\delta(x-x_{0})$. We show that the presence of the potential results in the…
We investigate the evolution of an electron undergoing coherent tunneling via adiabatic passage (CTAP) using the solution of the one-dimensional Schroedinger equation in both space and time for a triple well potential. We find the…
In this paper, the space-fractional Schr\"{o}dinger equations with singular potentials are studied. Delta-like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of 'weakened'…
We study the energy properties of a particle in one dimensional semi-harmonic rectangular wells and barriers. The integration of energies is obtained by solving a simple transcendental equation. Scattering states are shown to include cases…
We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $\Omega$ for any $t\ge T_\star,$ for some $T_\star>0)$ for solutions of the nonlinear Schr\"{o}dinger problem ${\rm i} u_t+\Delta u+a|u|^{-(1-m)}u=f(t,x),$ on a…
Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction boundary condition was derived. Here, we use the obtained Green's function to calculate…
We consider finite-time quantum quenches in the interacting Tomonaga--Luttinger model, for example time-dependent changes of the nearest-neighbour interactions for spinless fermions. We use the exact solutions for specific protocols…
A propagator for the one dimensional time-dependent Schr\"odinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and…
The effect of partial absorption on a diffusive particle which stochastically resets its position with a finite rate $r$ is considered. The particle is absorbed by a target at the origin with absorption `velocity' $a$; as the velocity $a$…
It has been found a simple procedure for the general solution of the time-independent Schr\"odinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not…
We introduce and solve from first principles a continuous-time quantum walk with absorption generated by a Lindblad boundary sink of arbitrary strength. Tracing out the sink maps the problem onto a non-Hermitian tight-binding Hamiltonian…
We derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schroedinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher…
We consider single-particle quantum transport on parametrized complex networks. Based on general arguments regarding the spectrum of the corresponding Hamiltonian, we derive bounds for a measure of the global transport efficiency defined by…
We consider the 1d Schr\"odinger operator with decaying random potential, and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions which is based on the formulation by…
We obtain a complete series solution of stationary Schr\"odinger's equation in the general quantum systems. It is exact in the sense that any approximation means is not used, or that the whole corrections or contributions from all order…
A Gedanken experiment is described to explore a counter-intuitive property of quantum mechanics. A particle is placed in a one-dimensional infinite well. The barrier on one side of the well is suddenly removed and the chamber dramatically…
We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{\omega t}$ with $U(x)=2\delta (x-a)$ or $U(x)= 2\delta(x-a)-2\delta (x+a)$. The particle is initially…
The extension of the Dirac Delta distribution (DD) to the complex field is needed for dealing with the complex-energy solutions of the Schr\"odinger equation, typically when calculating their inner products. In quantum scattering theory the…