Related papers: Numerical approximation of the Schr\"odinger equat…
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schr\"odinger equation provides full knowledge of all dynamical quantities of the physical system. However…
We give a rigorous proof for the existence of a finite-energy, self-similar solution to the focusing cubic Schr\"odinger equation in three spatial dimensions. The proof is computer-assisted and relies on a fixed point argument that shows…
We propose a class of numerical methods for the nonlinear Schr\"odinger (NLS) equation that conserves mass and energy, is of arbitrarily high-order accuracy in space and time, and requires only the solution of a scalar algebraic equation…
We investigate the possibility to assist the numerically ill-posed calculation of spectral properties of interacting quantum systems in thermal equilibrium by extending the imaginary-time simulation to a finite Schwinger-Keldysh contour.…
In this paper we derive and analyse new exponential collocation methods to efficiently solve the cubic Schr\"{o}dinger Cauchy problem on a $d$-dimensional torus. Energy preservation is a key feature of the cubic Schr\"{o}dinger equation. It…
The semiclassical Schr\"odinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to…
We consider the expansion of wave packets governed by the free Schr\"odinger equation. This seemingly simple task plays an important role in simulations of various quantum experiments and in particular in the field of matter-wave…
Analytical solutions of the N-dimensional Schr\"odinger equation for the newly proposed Varshni-Hulth\'en potential are obtained within the framework of Nikiforov-Uvarov method by using Greene-Aldrich approximation scheme to the centrifugal…
Analytical solutions are presented for eigenvalues, eigenfunctions of {\color{red} D-dimensional Schrodinger equation having Eckart potential} within Nikiforov-Uvarov method. This uses a new, improved approximation for centrifugal term,…
The real-time contour formalism for Green's functions provides time-dependent information of quantum many-body systems. In practice, the long-time simulation of systems with a wide range of energy scales is challenging due to both the…
We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the…
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known…
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…
The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector…
The time dependent complex Schr\"odinger equation with cubic nonlinearity is solved by constructing differential quadrature algorithm based on sinc functions. Reduction to a coupled system of real equations enables to approach the space…
Many phenomena of strongly correlated materials are encapsulated in the Fermi-Hubbard model whose thermodynamical properties can be computed from its grand canonical potential according to standard procedures. In general, there is no closed…
Finding accurate solutions to the Schr\"odinger equation is the key unsolved challenge of computational chemistry. Given its importance for the development of new chemical compounds, decades of research have been dedicated to this problem,…
In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical…
We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on…
The discrete Fourier transform is approximated by summing over part of the terms with corresponding weights. The approximation reduces significantly the requirement for computer memory storage and enhances the numerical computation…