In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number nj of the bound states and the sum of the phase shifts ηj (±M ) of the scattering states with the angular momentum j:(nj + 1)π when a half bound state occurs at E = M and j = 3/2 or − 1/2 (nj + 1)π when a half bound state occurs at E = −M and j = 1/2 or − 3/2 nj π the rest cases.The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.

The stretching and bending vibrational spectrum and the intensities of infrared transitions in a tetrahedral molecule are studied in two boson-realization models, where the interactions between stretching and bending vibrations are described by a quadratic cross term and by Fermi resonance terms, called harmonically coupled and Fermi resonance boson-realization model, respectively. The later is a development of our recent model. As an example, the two models are applied to the overtone spectrum and the intensities of silicon tetrafluorde. Those models provide fits to the published experimental vibrational eigenvalues with standard deviations 1.956 cm &1 and 0.908 cm &1 , respectively. The intensities of infrared transitions of its complete vibrations are calculated in the two models, and results show a good agreement with the observed data.1998 Academic Press

In terms of the modified Sturm-Liouville theorem, the two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential V(r) is established for an angular momentum m as a relation between the numbers n m Ϯ of the particle and antiparticle bound states and the phase shifts m (ϮM ): m ͑M͒Ϫ m ͑ϪM͒ϭ ͭ ͑n m ϩ Ϫn m Ϫ ϩ1͒ when a half-bound state occurs at EϭM for mϭ1 ͑ n m ϩ Ϫn m Ϫ Ϫ1 ͒ when a half-bound state occurs at EϭϪM for mϭ1 ͑ n m ϩ Ϫn m Ϫ ͒ the remaining cases. A solution of the Klein-Gordon equation with the energy M or ϪM is called a half-bound state if it is finite but does not decay fast enough at infinity to be square integrable.

No abstract