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We discuss results obtained recently for a quantum-mechanical model given by a neutral particle with a magnetic quadrupole moment in a radial electric field and a scalar potential proportional to the radial variable in cylindrical coordinates that also includes the noninertial effects of a rotating reference frame. We show that the conjectured allowed values of the cyclotron frequency are a mere artifact of the truncation of the power series used to solve the radial eigenvalue equation. Our analysis proves that the analytical expression for the eigenvalues is incorrect. [ABSTRACT FROM AUTHOR]

In this paper, a direct rigorous mathematical proof of the Gregory–Laflamme instability for the five-dimensional Schwarzschild black string is presented. Under a choice of ansatz for the perturbation and a gauge choice, the linearized vacuum Einstein equation reduces to an ordinary differential equation (ODE) problem for a single function. In this work, a suitable rescaling and change of variables is applied, which casts the ODE into a Schrödinger eigenvalue equation to which an energy functional is assigned. It is then shown by direct variational methods that the lowest eigenfunction gives rise to an exponentially growing mode solution, which has admissible behavior at the future event horizon and spacelike infinity. After the addition of a pure gauge solution, this gives rise to a regular exponentially growing mode solution of the linearized vacuum Einstein equation in harmonic/transverse-traceless gauge. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Using an expanded algebraic formalism with the inclusion of inverse operators, we construct raised and decreased coherent states for a set of exactly solvable quantum confined systems. We assume in this procedure both the ladder-operator and the displacement-operator methods, showing the equivalence between the two approaches. For each coherent state defined, we present its expansion in the Hilbert eigenstate space H e s , eigenvalue equation, overcompleteness relation, as well as other intrinsic properties. Whenever possible, we present an interpretation based on nonlinear deformation models for these new forms of coherent states. We evaluate the relevance of the new coherent states in quantum entanglement and squeezing by taking, as an example, the case of a coupled system. [ABSTRACT FROM AUTHOR]

Fractional quantum mechanics in the non-relativistic case is described by the space-fractional Schrödinger equation containing the fractional Riesz operator, which is a non-local operator. The non-locality makes it difficult to solve the fractional Schrödinger equation in the local potentials. In this paper, we study the solution of the fractional Schrödinger equation of a free particle moving on a circle by use of the Lévy path integrals approach. We present the Lévy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case. In addition, the Laplace transform, energy-time transform, and momentum representation of the free particle kernel are also obtained. The results of this paper contain the ones in standard quantum mechanics as special cases. The problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems, and the method of Lévy path integrals can be generalized to study more complex local potentials. [ABSTRACT FROM AUTHOR]

For a given discrete subgroup Γ of (ℂ, +) and given real number ν> 0, we study the spectral properties of the magnetic Laplacian operator Δν acting on the Hilbert space LΓ,χ2,v(ℂ) of (L2, Γ)-automorphic functions (see below for notations). We show that its spectrum is reduced to the eigenvalues νm; m= 0, 1, . . . . We also give a concrete description of each eigenspace in terms of the Hermite polynomials. This description will be used to characterize the range of true Bargmann transform of L2-periodic functions on the real line ℝ. [ABSTRACT FROM AUTHOR]

We consider a three-body one-dimensional Schrödinger operator with zero range potentials, which models a positive impurity with charge k > 0 interacting with an exciton. We study the existence of discrete eigenvalues as k is varied. On one hand, we show that for sufficiently small k there exists a unique bound state whose binding energy behaves like k4, and we explicitly compute its leading coefficient. On the other hand, if k is larger than some critical value, then the system has no bound states. [ABSTRACT FROM AUTHOR]

We show that the two-dimensional quantum-mechanical model proposed by Kulkarni and Pathak [J. Math. Phys. 62, 102102 (2021)] is separable. Consequently, instead of solving a two-dimensional Schrödinger equation, it is sufficient to solve two one-dimensional eigenvalue equations, one of which is exactly solvable. The solution to the remaining equation can be given in terms of a transcendental equation. [ABSTRACT FROM AUTHOR]

We prove results about the Hölder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive at a unitary analogue of the results of Damanik, Killip, and Lenz ["Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity," Commun. Math. Phys. 212, 191-204 (2000)] about the spectral measure of the discrete Schrödinger operator. [ABSTRACT FROM AUTHOR]

The eigenvalue problem for radial potentials is considered in a space whose spatial coordinates satisfy the SU(2) Lie algebra. As the consequence, the space has a lattice nature and the maximum value of momentum is bounded from above. The model shows interesting features due to the bound, namely, a repulsive potential can develop bound-states, or an attractive region may be forbidden for particles to propagate with higher energies. The exact radial eigen-functions in momentum space are given by means of the associated Chebyshev functions. For the radial stepwise potentials, the exact energy condition and the eigen-functions are presented. For a general radial potential, it is shown that the discrete energy spectrum can be obtained in desired accuracy by means of given forms of continued fractions. [ABSTRACT FROM AUTHOR]