Ground State Eigenfunction Research Articles

The electron-centric approach to the exchange-correlation energy.

The Kohn-Sham theory addresses the challenge of representing the kinetic energy by re-quantizing density functional theory at a level of non-interacting electrons. It transforms the many-electron problem into a fictitious non-interacting electron problem, with the many-electron effects concealed within the exchange-correlation (XC) energy, which is expressed in terms of the electron density ρ(r). Unlike the wave function, ρ(r) can be viewed as a classical quantity, and expressing the XC energy in terms of it circumvents the need for correlated wave functions. In this work, we once again employ the re-quantization strategy and determine the XC energy using a local one-particle Schrödinger equation. The ground-state eigenfunction of the corresponding Hamiltonian is a reference point (r) dependent orbital φr,σ(u, σ') which is subsequently used to generate the XC hole and the XC energy. The spin coordinate is denoted by σ and u is the electron-electron separation. The one-particle equation for φr,σ(u, σ') includes a local potential vr,σ(u, σ') that we approximate using two simple physical constraints. We assess the approximation by applying it to the helium iso-electronic series, the homogeneous electron gas, and the dissociation of the hydrogen molecule.

Ground state wavefunctions of elliptic relativistic integrable Hamiltonians

We derive ground state eigenfunctions and eigenvalues of various relativistic elliptic integrable models. The models we discuss appear in computations of superconformal indices of four-dimensional theories obtained by compactifying six-dimensional models on Riemann surfaces. These include, among others, the Ruijsenaars-Schneider model and the van Diejen model. The derivation of the eigenfunctions builds on physical inputs, such as conjectured Lagrangian across dimensions IR dualities and assumptions about the behavior of the indices in the limit of compactifications on surfaces with large genus/number of punctures/flux.

The temperature dependence on the coherence time of a quantum pseudodot qubit

In this work, the energies and eigenfunctions of ground state and first-excited states (GFES) of a strongly coupled polaron in a quantum pseudo-dot (QPD) were studied by using variational method of Pekar type (VMPT). A single qubit can be realized in this two-level quantum system. Then, we calculated the coherence time of a QPD qubit by employing the Fermi Golden Rule. The temperature effects on the coherence time are taken into account by using the quantum statistics theory (QST) and self-consistent calculation method. According to the obtained results, it is found that the coherence time increases with decreasing temperature. Also, this time is a decaying function of the chemical potential of the two-dimensional electron gas and the zero point of the PHP.

Analytical approach to the core-halo structure of fuzzy dark matter

Ultralight bosonic dark matter called fuzzy dark matter (FDM) has attracted much attention as an alternative to the cold dark matter. An intriguing feature of the FDM model is the presence of a soliton core, a stable dense core formed at the center of halos. In this paper, we analytically study the dependence of the soliton core properties on the halo characteristics by solving approximately the Schr\"odinger-Poisson equation. Focusing on the ground-state eigenfunction, we derive a key expression for the soliton core radius, from which we obtain the core-halo mass relations similar to those found in the early numerical work, but involving the factor dependent crucially on the halo concentration and cosmological parameters. Based on the new relations, we find that for a given cosmology, (i) there exist a theoretical bound on the radius and mass of soliton core for each halo mass (ii) incorporating the concentration-halo mass (C-M) relation into the predictions, the core-halo relations generally exhibit a non power-law behavior, and with the C-M relation suppressed at the low-mass scales, relevant to the FDM model, predictions tend to match the simulations well (iii) the scatter in the C-M relation produces a sizable dispersion in the core-halo relations, and can explain the results obtained from cosmological simulations. Finally, the validity of our analytical treatment are critically examined. A perturbative estimation suggests that the prediction of the core-halo relations is valid over a wide range of parameter space, and the impact of the approximation invoked in the analytical calculations is small, although it is not entirely negligible.

Superharmonic Double-well Systems with Zero-energy Ground States: Relevance for Diffusive Relaxation Scenarios

Relaxation properties (specifically time-rates) of the Smoluchowski diffusion process on a line, in a confining potential $ U(x) \sim x^m$, $m=2n \geq 2$, can be spectrally quantified by means of the affiliated Schr\"{o}dinger semigroup $\exp (-t\hat{H})$, $t\geq 0$. The inferred (dimensionally rescaled) motion generator $\hat{H}= - \Delta + {\cal{V}}(x)$ involves a potential function ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}$, $a=a(m), b=b(m) >0$, which for $m>2$ has a conspicuous higher degree (superharmonic) double-well form. For each value of $m>2$, $ \hat{H}$ has the zero-energy ground state eigenfunction $\rho _*^{1/2}(x)$, where $\rho _*(x) \sim \exp -[U(x)]$ stands for the Boltzmann equilibrium pdf of the diffusion process. A peculiarity of $\hat{H}$ is that it refers to a family of quasi-exactly solvable Schr\"{o}dinger-type systems, whose spectral data are either residual or analytically unavailable. As well, no numerically assisted procedures have been developed to this end. Except for the ground state zero eigenvalue and incidental trial-error outcomes, lowest positive energy levels (and energy gaps) of $\hat{H}$ are unknown. To overcome this obstacle, we develop a computer-assisted procedure to recover an approximate spectral solution of $\hat{H}$ for $m>2$. This task is accomplished for the relaxation-relevant low part of the spectrum. By admitting larger values of $m$ (up to $m=104$), we examine the spectral "closeness" of $\hat{H}$, $m\gg 2$ on $R$ and the Neumann Laplacian $\Delta _{\cal{N}}$ in the interval $[-1,1]$, known to generate the Brownian motion with two-sided reflection.

Minimum-uncertainty coherent states of the hyperbolic and trigonometric Rosen\u2013Morse oscillators

A mixed supersymmetric-algebraic approach is employed to generate the minimum uncertainty coherent states of the hyperbolic and trigonometric Rosen–Morse oscillators. The method proposed produces the superpotentials, ground state eigenfunctions and associated eigenvalues as well as the Schrödinger equation in the factorized form amenable to direct treatment in the algebraic or supersymmetric scheme. In the standard approach the superpotentials are calculated by solution of the Riccati equation for the given form of potential energy function or by differentiation of the ground state eigenfunction. The procedure applied is general and permits derivation the exact analytical solutions and coherent states for the most important model oscillators employed in molecular quantum chemistry, coherent spectroscopy (femtochemistry) and coherent nonlinear optics.

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. I. Generalities, cubic anharmonicity case

For the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text] the perturbation theory (PT) in powers of coupling constant [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) is developed constructively in [Formula: see text]-space and in [Formula: see text]-space, respectively. The Riccati–Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate [Formula: see text]-space and in [Formula: see text]-space, respectively, exploring the logarithmic derivative of wave function [Formula: see text]. It is shown that PT in powers of [Formula: see text] developed in RB equation leads to Taylor expansion of [Formula: see text] at small [Formula: see text] while being developed in GB equation leads to a new form of semiclassical expansion at large [Formula: see text]: it coincides with loop expansion in path integral formalism. In complementary way PT for large [Formula: see text] developed in RB equation leads to an expansion of [Formula: see text] at large [Formula: see text] and developed in GB equation leads to an expansion at small [Formula: see text]. Interpolating all four expansions for [Formula: see text] leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at [Formula: see text] for any coupling constant [Formula: see text] and dimension [Formula: see text]. As a concrete application, the low-lying states of the cubic anharmonic oscillator [Formula: see text] are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is [Formula: see text] for [Formula: see text] for coupling constant [Formula: see text] and dimension [Formula: see text] In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7–8 s.d. for [Formula: see text] and [Formula: see text]

On the ground state for quantum graphs

Ground-state eigenfunctions of Schrödinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs—differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices—a new class of vertex conditions introduced in the paper. It is shown that this class of vertex conditions is optimal. Relations to positivity preserving and positivity improving semigroups are clarified.

Nondegeneracy and stability of antiperiodic bound states for fractional nonlinear Schrödinger equations

We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schrödinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schrödinger operators on R with real-valued, periodic potentials as well as a Sturm–Liouville type oscillation theory for the higher antiperiodic eigenfunctions.

Spectral algorithms for multiple scale localized eigenfunctions in infinitely long, slightly bent quantum waveguides

A “bent waveguide” in the sense used here is a small perturbation of a two-dimensional rectangular strip which is infinitely long in the down-channel direction and has a finite, constant width in the cross-channel coordinate. The goal is to calculate the smallest (“ground state”) eigenvalue of the stationary Schrödinger equation which here is a two-dimensional Helmholtz equation, ψxx+ψyy+Eψ=0 where E is the eigenvalue and homogeneous Dirichlet boundary conditions are imposed on the walls of the waveguide. Perturbation theory gives a good description when the “bending strength” parameter ϵ is small as described in our previous article (Amore et al., 2017) and other works cited therein. However, such series are asymptotic, and it is often impractical to calculate more than a handful of terms. It is therefore useful to develop numerical methods for the perturbed strip to cover intermediate ϵ where the perturbation series may be inaccurate and also to check the pertubation expansion when ϵ is small. The perturbation-induced change-in-eigenvalue, δ≡E(ϵ)−E(0), is O(ϵ2). We show that the computation becomes very challenging as ϵ→0 because (i) the ground state eigenfunction varies on both O(1) and O(1∕ϵ) length scales and (ii) high accuracy is needed to compute several correct digits in δ, which is itself small compared to the eigenvalue E. The multiple length scales are not geographically separate, but rather are inextricably commingled in the neighborhood of the boundary deformation. We show that coordinate mapping and immersed boundary strategies both reduce the computational domain to the uniform strip, allowing application of pseudospectral methods on tensor product grids with tensor product basis functions. We compared different basis sets; Chebyshev polynomials are best in the cross-channel direction. However, sine functions generate rather accurate analytical approximations with just a single basis function.In the down-channel coordinate, X∈[−∞,∞], Fourier domain truncation using the change of coordinate X=sinh(Lt) is considerably more efficient than rational Chebyshev functions TBn(X;L). All the spectral methods, however, yielded the required accuracy on a desktop computer.

Influence of broken-pair excitations on the exact pair wavefunction

ABSTRACTDoubly occupied configuration interaction (DOCI), the exact diagonalisation of the Hamiltonian in the paired (seniority zero) sector of the Hilbert space, is a combinatorial cost wave function that can be very efficiently approximated by pair coupled cluster doubles (pCCD) at mean-field computational cost. As such, it is a very interesting candidate as a starting point for building the full configuration interaction (FCI) ground state eigenfunction belonging to all (not just paired) seniority sectors. The true seniority zero sector of FCI (referred to here as FCI0) includes the effect of coupling between all seniority sectors rather than just seniority zero, and is, in principle, different from DOCI. We here study the accuracy with which DOCI approximates FCI0. Using a set of small Hubbard lattices, where FCI is possible, we show that DOCI ∼ FCI0 under weak correlation. However, in the strong correlation regime, the nature of the FCI0 wavefunction can change significantly, rendering DOCI and pCCD a less than ideal starting point for approximating FCI.

Three Localized f Electrons in UPd2Al3and in UGe2Intermetallics

We have calculated the strength of the excitations between the crystal-field states which are in agreement with inelastic-neutron-scattering results. This agreement confirms the existence in the heavy-fermion superconductor UPd2Al3 the crystal-field electronic structure being the finger-print of the U ions with three localized f electrons forming strongly-correlated atomic-like quantum system 5f. The ionic integrity and the low-energy crystal-field electronic structure is preserved in this metallic system in the meV scale as has been postulated in the Quantum Atomistic Solid State theory (QUASST). We provide preliminary results with the U ion in UGe2 showing the ground-state eigenfunction which reproduces the ordered magnetic-moment value of 1.48 μB. This moment is composed from the dominant orbital contribution (2.6 μB) and the opposite spin moment (1.12 μB).

Pseudospectral method of solution of the Schrödinger equation with non classical polynomials; the Morse and Pöschl–Teller (SUSY) potentials

The eigenvalue spectra of the Schrödinger equation for the Morse and Pöschl–Teller potentials are calculated with three pseudospectral methods. The first method used is based on quadratures defined in terms of nonclassical polynomials orthogonal with a weight function equal to the square of the ground state eigenfunction. This is possible since the Morse and Pöschl–Teller potentials belong to the class of potentials in SUperSYmetric (SUSY) quantum mechanics. The other methods used are a Fourier grid method, often referred to as the discrete variable representation, and the Lagrange mesh method based on the Hermite quadrature. The convergence of the eigenvalues towards the known exact values versus the number of grid points is reported with the three methods for the Morse and Pöschl–Teller potentials. The difficulty to obtain converged results for the highest bound state is discussed.

Diffusion Monte Carlo study on temporal evolution of entropy and free energy in nonequilibrium processes.

A computational scheme to describe the temporal evolution of thermodynamic functions in stochastic nonequilibrium processes of isothermal classical systems is proposed on the basis of overdamped Langevin equation under given potential and temperature. In this scheme the associated Fokker-Planck-Smoluchowski equation for the probability density function is transformed into the imaginary-time Schrödinger equation with an effective Hamiltonian. The propagator for the time-dependent wave function is expressed in the framework of the path integral formalism, which can thus represent the dynamical behaviors of nonequilibrium molecular systems such as those conformational changes observed in protein folding and ligand docking. The present study then employs the diffusion Monte Carlo method to efficiently simulate the relaxation dynamics of wave function in terms of random walker distribution, which in the long-time limit reduces to the ground-state eigenfunction corresponding to the equilibrium Boltzmann distribution. Utilizing this classical-quantum correspondence, we can describe the relaxation processes of thermodynamic functions as an approach to the equilibrium state with the lowest free energy. Performing illustrative calculations for some prototypical model potentials, the temporal evolutions of enthalpy, entropy, and free energy of the classical systems are explicitly demonstrated. When the walkers initially start from a localized configuration in one- or two-dimensional harmonic or double well potential, the increase of entropy usually dominates the relaxation dynamics toward the equilibrium state. However, when they start from a broadened initial distribution or go into a steep valley of potential, the dynamics are driven by the decrease of enthalpy, thus causing the decrease of entropy associated with the spatial localization. In the cases of one- and two-dimensional asymmetric double well potentials with two minimal points and an energy barrier between them, we observe a nonequilibrium behavior that the system entropy first increases with the broadening of the initially localized walker distribution and then it begins to decrease along with the trapping at the global minimum of the potential, thus leading to the minimization of the free energy.

On the non-relativistic two-dimensional purely magnetic supersymmetric Pauli operator

The complete manifold of ground-state eigenfunctions for the purely magnetic two-dimensional Pauli operator is considered as a byproduct of a new reduction (found by the authors several years ago) for the algebro-geometric inverse spectral data (that is, Riemann surfaces and divisors). This reduction is associated with a -soliton hierarchy containing a 2D analogue of the famous `Burgers system'. This paper also surveys previous papers since 1980, including the first topological ideas in the space of quasi-momenta, and presents new results on self-adjoint boundary-value problems for the Pauli operator. The `non-spectral' Bloch–Floquet functions of zero 2D level give discrete points of additional spectrum analogous to the `boundary states' of finite-gap 1D potentials in the gaps. Bibliography: 35 titles.

F4 Quantum Integrable, rational and trigonometric models: space-of-orbits view

Algebraic-rational nature of the four-dimensional, F4-invariant integrable quantum Hamiltonians, both rational and trigonometric, is revealed and reviewed. It was shown that being written in F4 Weyl invariants, polynomial and exponential, respectively, both similarity-transformed Hamiltonians are in algebraic form, they are quite similar the second order differential operators with polynomial coefficients; the flat metric in the Laplace-Beltrami operator has polynomial (in invariants) matrix elements. Their potentials are calculated for the first time: they are meromorphic (rational) functions with singularities at the boundaries of the configuration space. Ground state eigenfunctions are algebraic functions in a form of polynomials in some degrees. Both Hamiltonians preserve the same infinite flag of polynomial spaces with characteristic vector (1, 2, 2, 3), it manifests exact solvability. A particular integral common for both models is derived. The first polynomial eigenfunctions are presented explicitly.

Two-Level Discretization Techniques for Ground State Computations of Bose-Einstein Condensates

This work presents a new methodology for computing ground states of Bose--Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a preprocessing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties. The nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without significant loss of accuracy when compared with the solution of the full fine scale problem. The preprocessing step is independent of the types and numbers of bosons. A postprocessing step further improves the accuracy of the method. We present rigorous a priori error estimates that predict convergence rates $H^3$ for the ground state eigenfunction and $H^4$ for the corresponding eigenvalue without pre-asymptotic effects; $H$ being the coarse scale discretization parameter. Numerical experiments indicate that these high rates may still be pessimistic.

Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

We prove the uniform lower bound for the difference $\lambda_2 - \lambda_1$ between first two eigenvalues of the fractional Schr\"odinger operator, which is related to the Feynman-Kac semigroup of the symmetric $\alpha$-stable process killed upon leaving open interval $(a,b) \in \R $ with symmetric differentiable single-well potential $V$ in the interval $(a,b)$, $\alpha \in (1,2)$. "Uniform" means that the positive constant appearing in our estimate $\lambda_2 - \lambda_1 \geq C_{\alpha} (b-a)^{-\alpha}$ is independent of the potential $V$. In general case of $\alpha \in (0,2)$, we also find uniform lower bound for the difference $\lambda_{*} - \lambda_1$, where $\lambda_{*}$ denotes the smallest eigenvalue related to the antisymmetric eigenfunction $\phi_{*}$. We discuss some properties of the corresponding ground state eigenfunction $\phi_1$. In particular, we show that it is symmetric and unimodal in the interval $(a,b)$. One of our key argument used in proving the spectral gap lower bound is some integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey-Lemma.

Automated quantum conductance calculations using maximally-localised Wannier functions

A robust, user-friendly, and automated method to determine quantum conductance in quasi-one-dimensional systems is presented. The scheme relies upon an initial density-functional theory calculation in a specific geometry after which the ground-state eigenfunctions are transformed to a maximally-localised Wannier function (MLWF) basis. In this basis, our novel algorithms manipulate and partition the Hamiltonian for the calculation of coherent electronic transport properties within the Landauer–Buttiker formalism. Furthermore, we describe how short-ranged Hamiltonians in the MLWF basis can be combined to build model Hamiltonians of large (>10,000 atom) disordered systems without loss of accuracy. These automated algorithms have been implemented in the Wannier90 code (Mostofi et al., 2008) [1], which is interfaced to a number of electronic structure codes such as Quantum-ESPRESSO, AbInit, Wien2k, SIESTA and FLEUR. We apply our methods to an Al atomic chain with a Na defect, an axially heterostructured Si/Ge nanowire and to a spin-polarised defect on a zigzag graphene nanoribbon.

On the construction of coherent states of position dependent mass Schrödinger equation endowed with effective potential

In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrödinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrödinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.