Exact Eigenfunctions Research Articles

Accurate Simulation of Efimov Physics in Ultracold Atomic Gases with Realistic Three-Body Multichannel Interactions

We give a detailed and self-contained description of a recently developed theoretical and numerical method for the simulation of three identical bosonic alkali-metal atoms near a Feshbach resonance, where the Efimov effect is induced. The method is based on a direct construction of the off-shell two-body transition matrix from exact eigenfunctions of the embedded two-body Hamiltonians, obtained using realistic parameterizations of the interaction potentials which accurately reproduce the molecular energy levels. The transition matrix is then inserted into the appropriate three-body integral equations, which may be efficiently solved on a computer. We focus especially on the power of our method in including rigorously the effects of multichannel physics on the three-body problem, which are usually accounted for only by various approximations. We demonstrate the method for 7Li, where we recently showed that a correct inclusion of this multichannel physics resolves the long-standing disagreement between theory and experiment regarding the Efimovian three-body parameter. We analyze the Efimovian enhancement of the three-body recombination rate on both sides of the Feshbach resonance, revealing strong sensitivity to the spin structure of the model thus indicating the prevalence of three-body spin-exchange physics. Finally, we discuss an extension of our methodology to the calculation of three-body bound-state energies.

Foldy-Wouthuysen transformation and multiwave states of a graphene electron in external fields and free (2+1)-space

The relativistic Foldy-Wouthuysen transformation is used for an advanced description of planar graphene electrons in external fields and free (2+1)-space. It is shown that the initial Dirac equation should by based on the usual (4 × 4) Dirac matrices but not on the reduction of matrix dimensions and the use of (2 × 2) Pauli matrices. Nevertheless, the both approaches agree with the experimental data on graphene electrons in a uniform magnetic field. The pseudospin of graphene electrons is not the one-value spin and takes the values ±1/2. The exact Foldy-Wouthuysen Hamiltonian of a graphene electron in uniform and nonuniform magnetic fields is derived. The exact energy spectrum agreeing with the experiment and exact Foldy-Wouthuysen wave eigenfunctions are obtained. These eigenfunctions describe multiwave (structured) states in the (2+1)-space. It is proven that the Hermite-Gauss beams exist even in the free space. In the multiwave Hermite-Gauss states, graphene electrons acquire nonzero effective masses dependent on a quantum number and move with group velocities which are less than the Fermi velocity. Graphene electrons in a static electric field also can exist in the multiwave Hermite-Gauss states defining non-spreading coherent beams. These beams can be accelerated and decelerated.

Disorder-tunable entanglement at infinite temperature.

Emerging quantum technologies hold the promise of unravelling difficult problems ranging from condensed matter to high-energy physics while, at the same time, motivating the search for unprecedented phenomena in their setting. Here, we use a custom-built superconducting qubit ladder to realize non-thermalizing states with rich entanglement structures in the middle of the energy spectrum. Despite effectively forming an "infinite" temperature ensemble, these states robustly encode quantum information far from equilibrium, as we demonstrate by measuring the fidelity and entanglement entropy in the quench dynamics of the ladder. Our approach harnesses the recently proposed type of non-ergodic behavior known as "rainbow scar," which allows us to obtain analytically exact eigenfunctions whose ergodicity-breaking properties can be conveniently controlled by randomizing the couplings of the model without affecting their energy. The on-demand tunability of quantum correlations via disorder allows for in situ control over ergodicity breaking, and it provides a knob for designing exotic many-body states that defy thermalization.

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined [Formula: see text] discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non-homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined [Formula: see text] space, the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.

Asymptotic formulas of the eigenvalues for the linearization of a one-dimensional sinh-Poisson equation

We are concerned with a Neumann problem of a one-dimensional sinh-Poisson equation u′′+λsinhu=0for0<x<1,u′(0)=u′(1)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u''+\\lambda \\sinh u=0 &{} \ ext {for}\\ 0<x<1,\\\\ u'(0)=u'(1)=0, \\end{array}\\right. } \\end{aligned}$$\\end{document}where lambda >0 is a parameter. A complete bifurcation diagram of this problem is obtained. We also consider the linearized eigenvalue problem at every nontrivial solution u. We derive exact expressions of all the eigenvalues and eigenfunctions, using Jacobi elliptic functions and complete elliptic integrals. Then, we also derive asymptotic formulas of eigenvalues as lambda rightarrow 0. Exact eigenvalues and eigenfunctions for a Dirichlet problem are presented without proof. The main technical tool is an ODE technique.

A dive into spectral inference networks: improved algorithms for self-supervised learning of continuous spectral representations

We propose a self-supervising learning framework for finding the dominant eigenfunction-eigenvalue pairs of linear and self-adjoint operators. We represent target eigenfunctions with coordinate-based neural networks and employ the Fourier positional encodings to enable the approximation of high-frequency modes. We formulate a self-supervised training objective for spectral learning and propose a novel regularization mechanism to ensure that the network finds the exact eigenfunctions instead of a space spanned by the eigenfunctions. Furthermore, we investigate the effect of weight normalization as a mechanism to alleviate the risk of recovering linear dependent modes, allowing us to accurately recover a large number of eigenpairs. The effectiveness of our methods is demonstrated across a collection of representative benchmarks including both local and non-local diffusion operators, as well as high-dimensional time-series data from a video sequence. Our results indicate that the present algorithm can outperform competing approaches in terms of both approximation accuracy and computational cost.

Torus geometry eigenfunctions of an interacting multi-Landau-level Hamiltonian

A short-ranged, rotationally symmetric multi-Landau-level model Hamiltonian for strongly interacting electrons in a magnetic field was proposed [A. Anand et al, Phys. Rev. Lett. 126, 136601 (2021)] with the key feature that it allows exact many-body eigenfunctions on the disk not just for quasiholes but for all charged and neutral excitations of the entire Jain sequence filling fractions. We extend this to geometries without full rotational symmetry, namely the torus and cylinder geometries, and present their spectra. Exact diagonalization of the interaction on the torus produces the low-energy spectra at filling fraction $\nu=n/(2pn+1)$ that is identical, up to a topological $(2pn+1)$-fold multiplicity, to that of the integer quantum Hall spectra at $\nu=n$, for the incompressible state as well as all excitations. While the ansatz eigenfunctions in the disk geometry cannot be generalized to closed geometries such as torus or sphere, we show how to extend them to cylinder geometry. Meanwhile, we show eigenfunctions for charged excitations at filling fractions between $\frac{1}{3}$ and $\frac{2}{5}$ can be written on the torus and the spherical geometries.

Projection-based guaranteed [formula omitted] error bounds for finite element approximations of Laplace eigenfunctions

For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the L2 norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of the conforming finite element method, and has an optimal speed of convergence for the eigenfunctions with the worst regularity. The resulting error estimate bounds the distance of spaces of exact and approximate eigenfunctions and, hence, is robust even in the case of multiple and tightly clustered eigenvalues. The accuracy of the proposed bound is illustrated by numerical examples.

Generalized fractional of the extended Nikiforov–Uvarov method for heavy tetraquark masses spectra

In this paper, the new analytic–exact energy eigenvalues and eigenfunctions are obtained in the fractional forms by using the extended Nikiforov–Uvarov approach in which the heavy diquark systems are described. We have recalculated the mass spectra and fractional radial wave of heavy tetraquarks. The mass spectra are compared to the experimental data. The present results show a good agreement with the experimental data and are improved in comparison with other studies. We conclude that the fractional models play a good role in the heavy tetraquark masses.

Position-dependent mass Schrödinger particles in space-like screw dislocation: associated degeneracies and magnetic and Aharonov–Bohm flux fields effects

We consider non-relativistic position-dependent mass (PDM) Schödinger particles moving in an elastic medium with space-like screw dislocation. Within the cylindrical coordinates $$\left( r,\phi ,z\right) $$ , we study and report the effects of screw dislocation as well as PDM settings on the energy levels of some PDM-quantum mechanical systems. In so doing, we use a power-law type positive-valued dimensionless scalar multiplier $$ f(r)=\lambda r^{\sigma }$$ (which manifestly introduces the metaphoric notion of PDM-Schödinger particles). Next, we subject such PDM particles to magnetic and Aharonov–Bohm flux fields. We report exact or conditionally exact eigenvalues and eigenfunctions for $$V(r)=0$$ and $$V(r) =a+b\,r+c\,r^{2}$$ .

On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schrödinger Operator

On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schrödinger Operator

The truncated Coulomb potential revisited

We apply the Frobenius method to the Schrödinger equation with a truncated Coulomb potential. By means of the tree-term recurrence relation for the expansion coefficients we truncate the series and obtain exact eigenfunctions and eigenvalues. From a judicious arrangement of the exact eigenvalues we derive useful information about the whole spectrum of the problem and can obtain other eigenvalues by simple and straightforward interpolation.

Exact quantum-mechanical equations for particle beams

The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.

Quantum-tunneling transitions and exact statistical mechanics of bistable systems with parametrized Dikand\xe9\u2013Kofan\xe9 double-well potentials

We consider a one-dimensional system of interacting particles (which can be atoms, molecules, ions, etc.), in which particles are subjected to a bistable potential the double-well shape of which is tunable via a shape deformability parameter. Our objective is to examine the impact of shape deformability on the order of transition in quantum tunneling in the bistable system, and on the possible existence of exact solutions to the transfer-integral operator associated with the partition function of the system. The bistable potential is represented by a class composed of three families of parametrized double-well potentials, whose minima and barrier height can be tuned distinctly. It is found that the extra degree of freedom, introduced by the shape deformability parameter, favors a first-order transition in quantum tunneling, in addition to the second-order transition predicted with the phi ^4 model. This first-order transition in quantum tunneling, which is consistent with Chudnovsky’s conjecture of the influence of the shape of the potential barrier on the order of thermally assisted transitions in bistable systems, is shown to occur at a critical value of the shape-deformability parameter which is the same for the three families of parametrized double-well potentials. Concerning the statistical mechanics of the system, the associate partition function is mapped onto a spectral problem by means of the transfer-integral formalism. The condition that the partition function can be exactly integrable, is determined by a criterion enabling exact eigenvalues and eigenfunctions for the transfer-integral operator. Analytical expressions of some of these exact eigenvalues and eigenfunctions are given, and the corresponding ground-state wavefunctions are used to compute the probability density which is relevant for calculations of thermodynamic quantities such as the correlation functions and the correlation lengths.Graphic

A most misunderstood conditionally-solvable quantum-mechanical model

In this paper we show that several authors have derived wrong physical conclusions from a gross misunderstanding of the exact eigenvalues and eigenfunctions of a conditionally-solvable quantum-mechanical model. It consists of an eigenvalue equation with seemingly Coulomb, linear and harmonic terms. Here we compare the results derived by those authors with the actual eigenvalues of the models calculated by means of the Ritz variational method.

Asymptotic expansions of eigenvalues by both the Crouzeix–Raviart and enriched Crouzeix–Raviart elements

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix–Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart–Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart–Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.

Quantum States of a Relativistic Charged Particle in Crossed Electric and Magnetic Fields

The exact eigenvalues and eigenfunctions of a spinless relativistic charged particle are obtained here. The results are compared with those from classical mechanics.

On the conformable fractional E(5) critical point symmetry

A new category of critical point symmetries is introduced. It is suggested that an element of this category, associated with zeros of conformable fractional Bessel functions, be utilized to describe spectra of nuclei around the E(5) critical point. The exact eigenvalue and eigenfunction solutions of local fractional Bohr-Mottelson Hamiltonian (with infinite square well potential) are obtained. The evolution of the spectra of the fractional E(5) critical point in correspondence with the fractional derivative order is investigated. Using the fractional E(5) critical point, a satisfactory description of the energy levels is obtained in the 102Pd and 104Ru nuclei.

An ubiquitous three-term recurrence relation

We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions for particular values of a model parameter. From these solutions, some researchers have derived a variety of predictions such as allowed angular frequencies, allowed field intensities, and the like. We also solve the eigenvalue equation numerically by means of the variational Rayleigh–Ritz method and compare the resulting eigenvalues with those provided by the truncation condition. In this way, we prove that those physical predictions are merely artifacts of the truncation condition.

Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.