Narrow Results

In the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n+1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W). We also show that geometries admitting Fischer–Marsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer–Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

The effective mass Klein–Gordon equation in one dimension for the Woods–Saxon potential is solved by using the Nikiforov–Uvarov method. Energy eigenvalues and the corresponding eigenfunctions are computed. Results are also given for the constant mass case.

While the usual harmonic oscillator potential gives discrete energies in the nonrelativistic case, it does not however give genuine bound states in the relativistic case if the potential is treated in the usual way. In the present article, we have obtained the eigenfunctions of the Dirac oscillator in two spatial dimensions, adapting the prescription of Moshinsky.

In this study, the Klein–Gordon equation (KGE) is solved with the attractive radial potential using the Nikiforov–Uvarov-functional-analysis (NUFA) method in higher dimensions. By employing the Greene–Aldrich approximation scheme, the approximate bound state energy equations as well as the corresponding radial wave function are obtained in closed form. Also, the expression for the scattering phase shift is obtained in D-dimensions. The effects of the screening parameter and the total angular momentum quantum number on the bound state energy and the scattering states’ phase shift are also studied numerically and graphically at different dimensions. An interesting result of this study is the inter-dimensional degeneracy symmetry for scattering phase shift. Hence, this concept is applicable in the areas of nuclear and particle physics.

For one-dimensional power-like potentials $|x{|}^m$, $m>0$, the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for $m>2$ but below them for $m<2$. The ground state BSE as function of $m$ are of the same order of magnitude as the exact energies for linear $(m=1)$, quartic $(m=4)$ and sextic $(m=6)$ oscillators but their relative deviation grows with $m$, reaching the value 4 at $m=\infty$. For physically important cases $m=1,4,6$, for the 100th excited state BSE coincide with exact ones in 5–6 figures.

It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction$\gamma$ (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function $\left|\gamma \right|<1/2$ for all $m\ge 1$. It grows slowly with increasing $m$ for fixed quantum number $N$, while it decays with quantum number growth at fixed $m$. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of $10^{-9}\text{–}10^{-11}$ (and $10^{-6}$).

We construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.

The Neumann boundary value problem of the Helmholtz equation of a vibrating circular membrane embedded into a flat rigid baffle is solved. The membrane is excited asymmetrically and radiates acoustic waves into the half-space above the baffle. A set of elementary asymptotic equations for modal radiation self-impedance and mutual impedance is presented. The equations are necessary for numerical computations of the radiated active and reactive acoustic power including the acoustic attenuation. A few equations available in the literature are collected. All the missing equations have been obtained using the methods of analysis of contour integral and stationary phase. The presented equations cover a wide frequency band, with the exception of the lowest frequencies and the frequencies close to coincidence.

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.

This paper applies a Hamiltonian method to study analytically the stress distributions of anisotropic two-dimensional elasticity for arbitrary boundary conditions without beam assumptions. The homogenous solutions consist of the eigensolutions of the derogatory zero eigenvalues zero eigensolutions and that of the well-behaved nonzero eigenvalues. The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions. On the other hand, the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. Completed numerical examples are newly given to compare with established results.