On Modified Second Paine–de Hoog–Anderssen Boundary Value Problem

This article deals with a special case of the Sturm–Liouville boundary value problem (BVP), an eigenvalue problem characterized by the Sturm–Liouville differential operator with unknown spectra and the associated eigenfunctions. By examining the BVP in the Schrödinger form, we are interested in the problem where the corresponding invariant function takes the form of a reciprocal quadratic form. We call this BVP the modified second Paine–de Hoog–Anderssen (PdHA) problem. We estimate the lowest-order eigenvalue without solving the eigenvalue problem but by utilizing the localized landscape and effective potential functions instead. While for particular combinations of parameter values that the spectrum estimates exhibit a poor quality, the outcomes are generally acceptable although they overestimate the numerical computations. Qualitatively, the eigenvalue estimate is strikingly excellent, and the proposal can be adopted to other BVPs.

Shadow cast and quasinormal modes in f(R) gravity model inspired by Yang–Mills field

In this paper, the null geodesic equations are computed in [Formula: see text] space–time dimensions [Y. Younesizadeh, A. A. Ahmad, A. H. Ahmed, F. Younesizadeh, Ann. Phys. 420, 168246 (2020)] by using the concept of symmetries and Hamilton–Jacobi equation and Carter separable method. With these null geodesics in hand, we evaluate the celestial coordinates (x, y) and the radius [Formula: see text] of the BH shadow and represent it graphically. In addition, we have shown that the peak of this energy slowly shifts to lower frequencies and its height decreases with the increase in the YM magnetic charge ([Formula: see text]) values and decrease in the [Formula: see text] parameter ([Formula: see text]) values. In addition, we have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form. The most important result derived from this study is that the height of this potential increases with increase in the YM magnetic charge ([Formula: see text]) values. Then, we study the quasinormal modes (QNMs) of these 4D black holes. For this purpose, we use the WKB approximation method upto third-order corrections. We have shown the perturbation’s decay in corresponding diagrams when the YM magnetic charge ([Formula: see text]) values and the [Formula: see text] parameter ([Formula: see text]) values change.

MODIFIED WAVE EQUATION FOR SPINLESS PARTICLES AND ITS SOLUTIONS IN AN EXTERNAL MAGNETIC FIELD

The wave equation for spinless particles with the Lorentz violating term is considered. We formulate the third-order in derivatives wave equation leading to the modified dispersion relation. The first-order formalism is considered and the density matrix is obtained. The Schrödinger form of equations is presented and the quantum-mechanical Hamiltonian is found. Exact solutions of the wave equation are obtained for particles in the constant and uniform external magnetic field. The change of the synchrotron radiation radius due to quantum gravity corrections is calculated.

Schrödinger-Iike equations for spin measurements, states, and wave functions, based on the statistical approach of Guiasu

Guiasu employed a statistical estimation principle to derive time-independent Schrödinger equations for the position but, as is usual, not the spin of a particle. Here, on the other hand, this principle is used to obtain Schrödinger-like equations for the spin but not the position of a particle. Steady states are described by continuous probability distributions, obtained by information-theoretic arguments, over spin measurements, states, and wave functions. These distributions serve as weight functions for orthogonal polynomials. Associated "wave functions," products of the polynomials and the square root of the weight function, satisfy differential equations, reducing to time-independent Schrödinger form at the point corresponding to the fully mixed spin-1/2 state.

THE BETHE–SALPETER EQUATION FOR MANY-BODY SYSTEMS

The mathematical technicalities involved in reducing the Bethe–Salpeter equation to the Schrodinger form of a wave equation are simplified by means of a formal modification of the interaction process for systems of many particles. This also makes it possible to carry out the reduction in a unique manner. The consequences of the modification are clearly explained, and it is shown that for atomic systems numerical results remain unchanged to order α2. There would seem to be some grounds for believing that numerical results remain unchanged to all orders, but no formal proof of this has been achieved.

A new method of calculating the mean value of 1/ r 3 for keplerian systems in quantum mechanics

The usual method of calculating the diagonal matrix elements of an integral power of the radius r in an inverse square quantum system is that due to Waller. His procedure is based on the Schrödinger form of the theory, and utilizes in particular the generating series for the Langueree polynomials. For negative powers, Dirac's very elegant theory of "q-numbers," developed in these 'proceedings' during the early days of quantum mechanics, furnishes an interesting alternative method which appears to have been overlooked, and which we believe is easier. From his theory the following rule can be derived: Suppose that we desire the mean value (diagonal element) of 1/r s , where s is an integer greater than unity. We write down the experssion with A(α l + β l e - i x + γ l e i x ) s-2 A = 16π 4 m 2 z 2 e 4 /( l + ½) n 3 h 4 , α l = 4π 2 m z e 2 / h 2 l ( l +1), β l = 2π 2 m Z e 2 / h 2 ( l +½)( l +1)[1-( l +1) 2 / n 2 ]½, γ l = 2π 2 m z e 2 / h 2 l ( l +½)[1- l 2 / n 2 ]½.