1. RELATIVISTIC HARMONIC OSCILLATOR Li Zhifeng Wolfgang Lucha Franz F. Schoeberl University of Vienna

2. Introduction and Motivation Generalization of Schroedinger operator towards relativistic Hamiltonian H, “Spinless Salpeter Equation” It’s difficult to obtain the analytical solution of this equation. The relativistic harmonic oscillator problem with Hamiltonian. A semianalytical approach was presented to this problem by using a recurrence scheme and get a compact expression of eigenfunction.

3. Contents 1. The Relativistic Harmonic-Oscillator(RHO) 2. Analytical approach to the solution of RHO bound-state eigenfunction 3. Energy eigenvalues of the bound states 4. Summary and Conclusion

4. 1.The Relativistic Harmonic-Oscillator(RHO) Hamiltonian of RHO the eigenstates equation We do some transformations

5. Transformation We get In momentum-space with

6. 2. Analytical approach to the solution of RHO bound-state eigenfunction Introduce the radial wave function which satisfies a radial equation We have and

7. Reconstruction of We construct in form of Taylor-series expansion with the expansion coefficients The first three terms

8. Analysis of Using the Leibniz’s theorem we get the recurrence relation

9. here By inspection of the function we find

10. We obtain and for where So the decomposes into two parts

11. The expression of

12. The analytical expressions for

13. 3. Energy eigenvalues of the bound states. In order to fulfill the normalization condition the solution must vanish in the limit for the practical purposes, the infinite series has to be cut to a large number N

14. According to minimum-maximum principle, where has to be computed numerically 0 1.37608 1 3.18131 2 4.99255 3 6.80514 4 8.61823 According to minimum-maximum principle, where has to be computed numerically

15. Comparison of the values with

16. 4. Summary and Conclusions We use power series to express the reduced eigenfunctions of the Hamiltonian with expression coefficients which determined by the recurrence relation. And we hope to find a way to construct the complete analytic solution. hep-ph/0501268 Journal of Mathematical Physics 46, 1 (2005)

17. Thank you very much!