Bohr Hamiltonian of triaxial nuclei using Morse plus screened Kratzer potentials with the extended Nikiforov–Uvarov method

In this work, Bohr Hamiltonian is used to explain the behavior of triaxial nuclei. A new potential, called Morse plus screened Kratzer potential, has been developed for the [Formula: see text]-part with [Formula: see text] fixed at [Formula: see text]. The Extended Nikiforov–Uvarov method involving Confluent Heun functions is used to derive the wave function and energy expression. The electric quadrupole transition rates and energy spectrum of platinum [Formula: see text] are determined and compared with the experimental data and some theoretical results.

Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.

The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids

<p>Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.</p>

The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids

<p>Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.</p>

Dirac Equation on the Kerr–Newman Spacetime and Heun Functions

By employing a pseudoorthonormal coordinate-free approach, the Dirac equation for particles in the Kerr–Newman spacetime is separated into its radial and angular parts. In the massless case to which a special attention is given, the general Heun-type equations turn into their confluent form. We show how one recovers some results previously obtained in literature, by other means.

Quantum relativistic cosmology: Dynamical interpretation and tunneling universe

In this work, the wave functions associated to the quantum relativistic universe, which is described by the Wheeler–DeWitt equation, are obtained. Taking into account different kinds of energy density, namely, matter, radiation, vacuum, dark energy and quintessence, we discuss some aspects of the quantum dynamics. In all these cases, the wave functions of the quantum relativistic universe are given in terms of the triconfluent Heun functions. We investigate the expansion of the universe using these solutions and found that the asymptotic behavior for the scale factor is [Formula: see text] for whatever the form of energy density is. On the other hand, we analyze the behavior at early stages of the universe and found that [Formula: see text]. We also calculate and analyze the transmission coefficient through the effective potential barrier.

Exact solutions of the harmonic oscillator plus non-polynomial interaction

The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a x 2 + b x 2 /(1 + c x 2 ) ( a > 0, c > 0) are given by the confluent Heun functions H c ( α , β , γ , δ , η ; z ). The minimum value of the potential well is calculated as V min ( x ) = − ( a + | b | − 2 a | b | ) / c at x = ± [ ( | b | / a − 1 ) / c ] 1 / 2 (| b | > a ) for the double-well case ( b < 0). We illustrate the wave functions through varying the potential parameters a , b , c and show that they are pulled back to the origin when the potential parameter b increases for given values of a and c . However, we find that the wave peaks are concave to the origin as the parameter | b | is increased.

Analytical investigation of wave absorption by a rotating black hole analogue

Perturbations in a draining vortex can be described analytically in terms of confluent Heun functions. In the context of analogue models of gravity in ideal fluids, we investigate analytically the absorption length of waves in a draining bathtub, a rotating black hole analogue, using confluent Heun functions. We compare our analytical results with the corresponding numerical ones, obtaining excellent agreement.