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A given set of data-points in some feature space may be associated with a Schrodinger equation whose potential is determined by the data. This is known to lead to good clustering solutions. Here we extend this approach into a full-fledged…

We consider three different approaches to analyze the quantum mechanical problems in multi-well potentials: i) the standard matrix diagonalization technique in the basis sets of harmonic oscillator eigenfunctions or plain waves; ii) the…

This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schr\"odinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates…

Functional defects in periodic media confine waves - acoustic, electromagnetic, electronic, spin, etc. - in various dimensions, depending on the structure of the defect. While defects are usually modelled by a superlattice with a typical…

We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions,…

We consider dimensional reduction techniques for the Liouville-von Neumann equation for the evaluation of the expectation values in a mixed quantum system. In applications such as nuclear spin dynamics the main goal for simulations is being…

The discrete spectra of certain two-dimensional Schrodinger operators are numerically calculated. These operators have interesting spectral properties, i.e. their kernels are multi-dimensional and the deformations of potentials via the…

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets.…

We present a refinement of the Spectral Method by incorporating an optimization method into it and generalize it to two space dimensions. We then apply this Refined Spectral Method as an extremely accurate technique for finding the bound…

Resolvent analysis is a powerful tool for studying coherent structures in turbulent flows. However, its application beyond canonical flows with symmetries that can be used to simplify the problem to inherently three-dimensional flows and…

This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method…

This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…

Collective variable-based enhanced sampling methods are routinely used on systems with metastable states, where high free energy barriers impede proper sampling of the free energy landscapes when using conventional molecular dynamics…

An exactly solvable position-dependent mass Schr\"odinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems…

This research addresses the numerical simulation of the Boltzmann transport equation for semiconductor devices by proposing a multidimensional self-adaptive numerical simulation framework. This framework is applied to two important…

Basing on analogy between the three-body scattering problem and the diffraction problem of the plane wave (for the case of the short range pair potentials) by the system of six half transparent screens, we presented a new approach to the…

In a previous publication [J. Chem. Phys., 161, 044105 (2024)], it has been shown that Rothe's method can be used to solve the time-dependent Schr\"odinger equation (TDSE) for the hydrogen atom in a strong laser field using time-dependent…

We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schr\"odinger eigenvalue equation $H\Psi \equiv (\Delta_2 +V)\Psi=E\Psi$ on any 2D Riemannian…

This paper considers large-scale simulations of wave propagation phenomena. We argue that it is possible to accurately compute a wavefield by decomposing it onto a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen…

Specialized function gradient computing hardware could greatly improve the performance of state-of-the-art optimization algorithms, e.g., based on gradient descent or conjugate gradient methods that are at the core of control, machine…