On the Stronger Statement of Levinson's Theorem for the Dirac Equation

Recently a stronger statement of Levinson's theorem for the Dirac equation was presented, where the limits of the phase shifts at $E=\pm M$ are related to the numbers of nodes of radial functions at the same energies, respectively. However, in this letter we show that this statement has to be modified because the limits of the phase shifts may be negative for the Dirac equation

Exact solution to the Schr\"{o}dinger equation for the quantum rigid body

The exact solution to the Schr\"{o}dinger equation for the rigid body with the given angular momentum and parity is obtained. Since the quantum rigid body can be thought of as the simplest quantum three-body problem where the internal motion is frozen, this calculation method is a good starting point for solving the quantum three-body problems.Comment: 8 pages, no figure, LaTex, will be published in Foundations of Physics Letters, 12 (1999) 561-57

RR Matrix for UqE7U_{q}E_{7}

The quantum Clebsch-Gordan coefficients and the explicit form of the $\breve{R}_{q}$ matrix related with the minimal representation of the quantum enveloping algebra $U_{q}E_{7}$ are calculated in this paper

The (2+1) Dirac Equations with δ\delta Potential

In this Letter the bound states of (2+1) Dirac equation with the cylindrically symmetric $\delta (r-r_{0})$-potential are discussed. It is surprisingly found that the relation between the radial functions at two sides of $r_{0}$ can be established by an SO(2) transformation. We obtain a transcendental equation for calculating the energy of the bound state from the matching condition in the configuration space. The condition for existence of bound states is determined by the Sturm-Liouville theorem.Comment: Latex 11 pages accepted by Found. Phys. Let

Exact Solutions to the Schr\"{o}dinger Equation for the potential V(r)=ar2+br−4+cr−6V(r)=a r^2+b r^{-4}+c r^{-6} in 2D

Making use of an ${\it ansatz}$ for the eigenfunctions, we obtain an exact closed form solution to the non-relativistic Schr\"{o}dinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$ in two dimensions, where the parameters of the potential $a, b, c$ satisfy some constraints.Comment: Latex file, pages 9 and 2 eps figures, accepted by J. Phys.

The modified Seiberg-Witten monopole equations and their exact solutions

The modified Seiberg-Witten monopole equations are presented in this letter. These equations have analytic solutions in the whole 1+3 Minkowski space with finite energy. The physical meaning of the equations and solutions are discussed here.Comment: RevTex, 6 page, no figur

U(2) algebraic model applied to stretching vibrational spectra of tetrahedral molecules

The highly excited stretching vibrational energy levels and the intensities of infrared transitions in tetrahedral molecules are studied in a U(2) algebraic model. Its applications to silane and silicon tetrafluoride are presented with smaller standard deviations than those of other models.Comment: Revtex, 6 pages, no figure, to appear in Int. J. Theor. Phy

Exact Solutions to the Schr\"{o}dinger Equation for the Inverse-Power Potential in Two Dimensions

Utilizing an ${\it ansatz}$ for the eigenfunctions, we arrive at an exact closed form solution to the Schr\"{o}dinger equation with the inverse-power potential, $V(r)=ar^{-4}+br^{-3}+cr^{-2}+dr^{-1}$ in two dimensions, where the parameters of the potential $a, b, c, d$ satisfy a constraint.Comment: Latex file 9 pages and submit to Euro. Phys. J.

The rotational invariants constructed by the products of three spherical harmonic polynomials

The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors, where the coefficients are calculated explicitly in this paper

Thermally induced structural modification in the Al/Zr multilayers

The effect of increasing temperature on the structural stability and interactions of two kinds of Al/Zr (Al(1%wtSi)/Zr and Al(Pure)/Zr) multilayer mirrors are investigated. All Al/Zr multilayers annealed from 200^{\circ}C to 500^{\circ}C, were deposited on Si wafers by using direct-current magnetron sputtering technology. A detailed and consistent picture of the thermally induced changes in the microstructure is obtained using an array of complementary measurements including grazing incidence X-ray reflectance, atomic force microscope, X-ray diffraction and high-resolution transmission electron microscopy. The first significant structural changes of two systems are observed at 250^{\circ}C, characterized by asymmetric interlayers appears at interface. At 290^{\circ}C, the interface consisted of amorphous Al-Zr alloy is transformed to amorphous Al-Zr alloy and cubic ZrAl3 in both systems. By 298^{\circ}C of Al(1%wtSi)/Zr and 295^{\circ}C of Al(Pure)/Zr multilayers, the interfacial phases of Al-Zr alloy transform completely into polycrystalline mixtures of hcp-ZrAl2 and cubic-ZrAl3, which smooth the interface boundary and lower the surface roughness in the multilayers. Up to 500^{\circ}C, the multilayer structure still exists in both systems, and the differences between the asymmetric interlayers are much larger in the multilayers. Finally, we discuss the transformation from symmetric to asymmetric in the annealing process for other systems