Related papers: Gaussian wave packet transform based numerical sch…
This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena.…
We reexamine the general solution of a Schr\"{o}dinger equation in the presence of a time-dependent linear potential in configuration space based on the Lewis-Riesenfeld framework. For comparison, we also solve the problem in momentum space…
We show that the Schr\"{o}dinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$,…
This paper is devoted to the cubic nonlinear Schr\"odinger equation in a two dimensional waveguide with shrinking cross section of order $\epsilon$. For a Cauchy data living essentially on the first mode of the transverse Laplacian, we…
We present a quantum computational framework that systematically converts classical linear iterative algorithms with fixed iteration operators into their quantum counterparts using the Schr\"odingerization technique [Shi Jin, Nana Liu and…
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting…
This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schr\"odinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly…
In this paper, we consider the cubic nonlinear Schr\"odinger equation with third order dispersion on the circle. In the non-resonant case, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac 34$,…
We propose new methods designed to numerically approximate the solution to the time dependent Schr{\"o}dinger equation, based on two types of ansatz: tensors, and approximation by a linear combination of gaussian wave packets. In both…
In this article we discuss our ongoing program to extend the scope of certain, well-developed microlocal methods for the asymptotic solution of Schr\"{o}dinger's equation (for suitable `nonlinear oscillatory' quantum mechanical systems) to…
We consider a model dissipative quantum-mechanical system realized by coupling a quantum oscillator to a semi-infinite classical string which serves as a means of energy transfer from the oscillator to the infinity and thus plays the role…
We use the Wigner transformation and asymptotic analysis to systematically derive the semi-classical model for the Schr\"{o}dinger equation in arbitrary spatial dimensions, with any periodic structure. Our particular emphasis lies in…
We present quantum algorithms for simulating the dynamics of a broad class of classical oscillator systems containing $2^n$ coupled oscillators (Eg: $2^n$ masses coupled by springs), including those with time-dependent forces, time-varying…
We formulate the Schr\"odinger equation as the equation of motion of a small external influence which serves as the initial boundary condition of a physical system in classical laboratory space. The Hilbert space of possible external…
The aim of this paper is to study the semi-classical behaviour of Schr\"odinger's dynamics for an one-dimesional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an…
We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the…
We present quantum algorithms for electromagnetic fields governed by Maxwell's equations. The algorithms are based on the Schr\"odingersation approach, which transforms any linear PDEs and ODEs with non-unitary dynamics into a system…
We study numerically the dynamics of a one-electron wave packet in a two-dimensional random lattice with long-range correlated diagonal disorder in the presence of a uniform electric field. The time-dependent Schr\"{o}dinger equation is…
We study the partial data Calder\'on problem for the anisotropic Schr\"{o}dinger equation \begin{equation} \label{eq: a1} (-\Delta_{\widetilde{g}}+V)u=0\text{ in }\Omega\times (0,\infty), \end{equation} where $\Omega\subset\mathbb{R}^n$ is…
We modify the Schr\"{o}dinger equation in a way that preserves its main properties but makes use of higher order derivative terms. Although the modification represents an analogy to the Doebner-Goldin modification, it can differ from it…