Related papers: Reference potential approach to the quantum-mechan…
A recently proposed reference potential approach to the inverse Schr\"{o}dinger problem is further developed. As previously, theoretical developments are demonstrated on example of diatomic xenon molecule in its ground electronic state. An…
Elegant and mathematically rigorous methods of the quantum inverse theory are difficult to put into practice because there is always some lack of needful input information. In this situation, one may try to construct a reference potential,…
A complete solution to the inverse problem of Kohn-Sham (KS) density functional theory is proposed. Our method consists of two steps. First, the effective KS potential is determined from the ground state density of a given system. Then, the…
A novel theoretical framework, the inverse problem approach, is proposed to calculate non-perturbative quantities in quantum chromodynamics (QCD). Based on the dispersion relation of quantum field theory, this approach determines unknown…
In this work, we study the inverse problem of analog gravity systems which admit rotation and energy-dependent boundary conditions. By extending two recent results, we provide a recipe that allows one to relate resonant transmission spectra…
A technique to reconstruct one-dimensional, reflectionless potentials and the associated quantum wave functions starting from a finite number of known energy spectra is discussed. The method is demonstrated using spectra that scale like the…
We establish connections between different approaches to inverse spectral problems: the classical Gelfand--Levitan theory, the Krein method, the Simon theory, the approach proposed by Remling and the Boundary Control method. We show that…
We study the inverse dynamic problem of recoverying the potential in the one-dimensional dynamical system with memory. The Gelfand--Levitan equations are derived for the kernel of the integral operator which is inverse to the control…
We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic motion of a particle in the potential field $V(x)=g_{1}x^{-1}+g_{2}x^{-2}$. For $g_{2}>0$ and $g_{1}<0$, the potential is known as the…
Comparison of numerical performances of two methods for coefficient inverse problems is described. The first one is the classical Gel'fand-Levitan-Krein equation method, and the second one is the recently developed approximately globally…
Inverse scattering and spectral one-dimensional problems are discussed systematically in a self-contained way. Many novel results, due to the author are presented. The classical results are often presented in a new way. Several highlights…
Using a new Bayesian method for solving inverse quantum problems, potentials of quantum systems are reconstructed from coordinate measurements in non-stationary states. The approach is based on two basic inputs: 1. a likelihood model,…
A method for solving an inverse spectral problem for the one-dimensional Dirac equation is developed. The method is based on the Gelfand-Levitan equation and the Fourier-Legendre series expansion of the transmutation kernel. A linear…
We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is…
The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables, similarly as in the solution of the Vlasov-Poisson system by means of the Bernstein-Greene-Kruskal method. In the…
Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real function (potential), is solved. Closed system…
We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral…
The inverse problem of Kohn-Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the…
Sum of a second derivative operator with periodic boundary conditions and an integral operator of rank one (non-local potential) is studied in this manuscript. Not only spectral analysis is conducted for this operator but the inverse…
The inversion in the sphere or Kelvin transformation, which exchanges the radial coordinate for its inverse, is used as a guide to relate distinct electrostatic problems with dual features. The exact solution of some nontrivial problems are…