Stationary Schr Research Articles

Generalized Ellis-Bronnikov bilayer graphene wormholelike surface

In this paper, we investigate the spinless stationary Schr\"odinger equation for the electron when it is permanently bound to a generalized Ellis-Bronnikov bilayer graphene wormholelike surface. The curvature gives rise to a geometric potential affecting thus the electronic dynamics. The geometry of the wormhole's shape is controlled by the parameter $n$ which assumes even values. We discuss the role played by the parameter $n$ and the orbital angular momentum on bound states and probability density for the electron.

Numerov and phase-integral methods for charmonium

This paper applies the Numerov and phase-integral methods to the stationary Schrödinger equation that studies bound states of charm anti-charm quarks. The former is a numerical method well suited for a matrix form of the second-order ordinary differential equations, and can be applied whenever the stationary states admit a Taylor-series expansion. The latter is an analytic method that provides, in principle, even exact solutions of the stationary Schrödinger equation, and well suited for applying matched asymptotic expansions and higher-order quantization conditions. The Numerov method is found to be always in agreement with the early results of Eichten et al., whereas an original evaluation of the phase-integral quantization condition clarifies under which conditions the previous results in the literature on higher-order terms can be obtained.

Weighted nodal domain averages of eigenstates for quantum Monte Carlo and beyond

We study the nodal properties of many-body eigenstates of stationary Schr\"odinger equation that affect the accuracy of real-space quantum Monte Carlo calculations. In particular, we introduce weighted nodal domain averages that provide a new probe of nodal surfaces beyond the usual expectations. Particular choices for the weight function reveal, for example, that the difference between two arbitrary fermionic eigenvalues is given by the nodal hypersurface integrals normalized by overlaps with the bosonic ground state of the given Hamiltonian. Noninteracting and fully interacting Be atom with corresponding almost exact and approximate wave functions are used to illustrate several aspects of these concepts. Variational formulations that employ different weights are proposed for prospective improvement of nodes in variational and fixed-node diffusion Monte Carlo calculations.

Atomic Siegert states in a rotating electric field

We show that the time-dependent Schr\"odinger equation describing in the dipole approximation the interaction of a one-electron atom with a monochromatic circularly polarized electromagnetic field can be reduced to a stationary Schr\"odinger equation in the form which allows to separate variables in the asymptotic region and explicitly formulate outgoing-wave boundary conditions. The solutions to this equation are called atomic Siegert states (SSs) in a rotating electric field. We develop the theory of such states and propose an efficient method to construct them using powerful techniques of stationary scattering theory. The method yields not only the SS eigenvalue defining the Stark-shifted energy and total ionization rate of the state, but also the SS eigenfunction defining partial ionization amplitudes and the photoelectron momentum distribution. The theory is illustrated by calculations for a model potential.

Stability analysis on dark solitons in quasi-1D Bose\u2013Einstein condensate with three-body interactions

The stability properties of dark solitons in quasi-one-dimensional Bose–Einstein condensate (BEC) loaded in a Jacobian elliptic sine potential with three-body interactions are investigated theoretically. The solitons are obtained by the Newton-Conjugate Gradient method. A stationary cubic-quintic nonlinear Schrödinger equation is derived to describe the profiles of solitons via the multi-scale technique. It is found that the three-body interaction has distinct effect on the stability properties of solitons. Especially, such a nonlinear system supports the so-called dark solitons (kink or bubble), which can be excited not only in the gap, but also in the band. The bubbles are always linearly and dynamically unstable, and they cannot be excited if the three-body interaction is absent. Both stable and unstable kinks, depending on the physical parameters, can be excited in the BEC system.

Retraction Note: Boundary behaviors for linear systems of subsolutions of the stationary Schr\xf6dinger equation

This paper investigates the boundary behaviors for linear systems of subsolutions of the stationary Schrodinger equation, which contain unstable subsystems. Our first aim is to establish a state-feedback switching law guaranteeing the continuous-time systems to be uniformly exponentially stable. And then we present sufficient and necessary for the stability of the systems with two Schrodinger subsystems. Finally, an illustrative example is given to verify the result.

On a stationary Schrödinger equation with periodic magnetic potential

We prove existence results for a stationary Schr{\"o}dinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.

Retraction Note: Minimally thin sets associated with the stationary Schr\xf6dinger operator

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1186/s13660-021-02575-1

Research in the Field of Geometric Analysis at Volgograd State University

This article discusses the main directions of research in geometric analysis, which were conducted and are being carried out by the scientific mathematical school of Volgograd State University. The results of the founder of the scientific school, Doctor of Physics and Mathematics, Professor Vladimir Mikhailovich Miklyukov and his students are summarized. These results concern the solution of a number of problems in the field of quasiconformal flat mappings and mappings with bounded distortion of surfaces and Riemannian manifolds, the theory of minimal surfaces and surfaces of prescribed mean curvature, surfaces of zero mean curvature in Lorentz spaces, as well as problems associated with the study of the stability of such surfaces. In addition, the results of the study of various classes of triangulations — an object that appears at the junction of research in the field of geometric analysis and computational mathematics — are noted. Besides, this review discusses papers that use the Fourier decomposition method for solutions of the Laplace — Beltrami equations and the stationary Schr¨odinger equation with respect to the eigenfunctions of the corresponding boundary value problems. In particular, the authors give the results on finding capacitive characteristics that allowed for the first time to formulate and prove the criteria for the fulfillment of various theorems of Liouville type and the solvability of boundary value problems on model and quasimodel Riemannian manifolds. The role of the equivalent function method is also indicated in the study of such problems on manifolds of a fairly general form. In addition to this, the article gives an overview of the results concerning estimates of calculating error integral functionals and convergence of piecewise polynomial solutions of nonlinear variational type equations: minimal surface equations, equilibrium equations capillary surface and equations of biharmonic functions.

Light dressing of a diatomic superconducting artificial molecule

In this work, we irradiate a superconducting artificial molecule composed of two coupled tunable transmons with microwave light while monitoring its state via joint dispersive readout. Performing high-power spectroscopy, we observe and identify a variety of single- and multiphoton transitions. We also find that at certain fluxes, the measured spectrum of the system deviates significantly from the solution of the stationary Schr\"odinger equation with no driving. We reproduce these unusual spectral features by solving numerically the full master equation for a steady-state and attribute them to an Autler-Townes-like effect in which a single tone is simultaneously dressing the system and probing the transitions between new eigenstates. We show that it is possible to find analytically the exact frequencies at which the satellite spectral lines appear by solving self-consistent equations in the rotating frame. Our approach agrees well with both the experiment and the numerical simulation.

Ambiguity of applying the Wildermuth-Tang rule to estimate the quasibound states of α particles in α emitters

The Wildermuth-Tang (WT) prescription is used to verify the Bohr-Sommerfeld (BS) quantization condition in the $\ensuremath{\alpha}$-decay problem. It gives the global quantum number that relates the number of nodes of the quasibound radial wave function of the $\ensuremath{\alpha}$-daughter system to the shell model and Pauli exclusion principle. Here we examine the applicability of the WT rule in the $\ensuremath{\alpha}$-decay microscopic calculations that start with solving the stationary Schr\"odinger wave equation for different types of the interaction potentials. We found that applying the BS quantization condition along with the WT prescription for the potentials that have no internal pocket yields a large number of nodes in the radial wave function compared to the potentials characterized with an automatic physical internal pocket, which likely produce nodeless or at most a two-node interior wave function. This gives confidence in the latter type of the potentials that efficaciously simulates the Pauli principle by considering the change in the intrinsic kinetic energy. However, it is possible to reproduce the observed half-life data using the potentials that have no automatic internal pocket with applying the BS quantization condition with quantum numbers which are significantly less than that obtained from the WT rule, upon properly normalizing the potential.

Retraction Note: Boundary behaviors of modified Green\u2019s function with respect to the stationary Schr\xf6dinger operator and its applications

The Editors-in-Chief have retracted this article [1] because it significantly overlaps with a number of previously published articles from different authors [2–4]. The author has not responded to any correspondence regarding this retraction.

Gravitational atoms: General framework for the construction of multistate axially symmetric solutions of the Schrödinger-Poisson system

We present a general strategy to solve the stationary Schr\"odinger-Poisson (SP) system of equations for multistates with axial symmetry. The approach allows us to obtain the well known single and multistate solutions with spherical symmetry, Newtonian multistate $\ell-$boson stars and axially symmetric multistate configurations. For each case we construct particular examples that illustrate the method, whose stability properties are studied by numerically solving the time-dependent SP system. Among the stable configurations there are the mixed-two-state configurations including spherical and dipolar components, which might have an important value as potential anisotropic dark matter halos in the context of ultralight bosonic dark matter scenarios. This is the reason why we also present a possible process of formation of these mixed-two-state configurations that could open the door to the exploration of more general multistate structure formation scenarios.

Retraction Note: Fixed point theorems for solutions of the stationary Schr\xf6dinger equation on cones

The Editors-in-Chief have retracted this article [1] because the results presented are invalid. The article also shows significant overlap with a number of previously published articles [2–5] and evidence of both peer review and authorship manipulation.The authors have not responded to any correspondence regarding this retraction.

Stationary Schr\xf6dinger equation in the semi-classical limit: WKB-based scheme coupled to a turning point

This paper is concerned with the efficient numerical treatment of 1D stationary Schrödinger equations in the semi-classical limit when including a turning point of first order. As such it is an extension of the paper [3], where turning points still had to be excluded. For the considered scattering problems we show that the wave function asymptotically blows up at the turning point as the scaled Planck constant varepsilon rightarrow 0, which is a key challenge for the analysis. Assuming that the given potential is linear or quadratic in a small neighborhood of the turning point, the problem is analytically solvable on that subinterval in terms of Airy or parabolic cylinder functions, respectively. Away from the turning point, the analytical solution is coupled to a numerical solution that is based on a WKB-marching method—using a coarse grid even for highly oscillatory solutions. We provide an error analysis for the hybrid analytic-numerical problem up to the turning point (where the solution is asymptotically unbounded) and illustrate it in numerical experiments: if the phase of the problem is explicitly computable, the hybrid scheme is asymptotically correct w.r.t. varepsilon . If the phase is obtained with a quadrature rule of, e.g., order 4, then the spatial grid size has the limitation h=mathcal{O}(varepsilon ^{7/12}) which is slightly worse than the h=mathcal{O}(varepsilon ^{1/2}) restriction in the case without a turning point.

Schr\xf6dinger-type identity to the existence and uniqueness of a solution to the stationary Schr\xf6dinger equation

In this article we study applications of the Schrödinger-type identity for obtaining transmutations via the fixed point index for nonlinear integral equations. It is possible to derive a wide range of transmutation operators by this method. Classical Riesz transforms are involved in the Schrödinger-type identity method as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. In this paper, we present a modified Schrödinger-type identity for solutions of a class of linear Schrödinger equations with mixed boundary conditions. The techniques used in our proofs are quite different, and most remarkably some of the proofs become simpler and more straightforward. As an application, we obtain the existence and uniqueness of a solution to the stationary Schrödinger equation in the sense of the Weyl law, which advances the recent results obtained in several articles even in a more general setting.

Stationary Real Solutions of the Nonlinear Schrödinger Equation on a Ring with a Defect

We analyze the 1D cubic nonlinear stationary Schr\"odinger equation on a ring with a defect for both focusing and defocusing nonlinearity. All possible $\delta$ and $\delta'$ boundary conditions are considered at the defect, computing for each of them the real eigenfunctions, written as Jacobi elliptic functions, and eigenvalues for the ground state and first few excited energy levels. All six independent Jacobi elliptic functions are found to be solutions of some boundary condition. We also provide a way to map all eigenfunctions satisfying $\delta$/$\delta'$ conditions to any other general boundary condition or point-like potential.

Six-Vertex Model and Non-linear Differential Equations. I: Spectral Problem

In this work we relate the spectral problem of the toroidal six-vertex model's transfer matrix with the theory of integrable non-linear differential equations. More precisely, we establish an analogy between the Classical Inverse Scattering Method and previously proposed functional equations originating from the Yang-Baxter algebra. The latter equations are then regarded as an Auxiliary Linear Problem allowing us to show that the six-vertex model's spectrum solves Riccati-type non-linear differential equations. Generating functions of conserved quantities are expressed in terms of determinants and we also discuss a relation between our Riccati equations and a stationary Schr\"odinger equation.

Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity

We introduce an exactly integrable singular potential for which the solution of the one-dimensional stationary Schr\"odinger equation is written through irreducible linear combinations of the Gauss hypergeometric functions. The potential, which belongs to a general Heun family, is a short-range one that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. We derive the exact spectrum equation for the energy and discuss the bound states supported by the potential.

Solutions of nonlinear equations of divergence type in domains having corner points

The algorithm for generation of exact solutions of the nonlinear equation in partial derivatives of a divergent type which is included in the formulation of magnetostatics, hydro-and aerodynamics, quantum mechanics (stationary Schr\"odinger equation) has been suggested. The properties of smoothness of solutions in domains with corner points (piecewise smooth boundary) have been considered. The solutions with unlimited derivatives in the corner point domain have been presented on the basis of a new class of special functions.