1846

### An Expansion Method for Schrödinger Equation of Quantum Billiards with Arbitrary Shapes

**Abstract:**A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional
axisymmetric region is developed. The boundary of the region
is defined by an arbitrary analytic function. The method uses a
coordinate transformation and an expansion in eigenfunctions. The
effectiveness is checked and confirmed by applying the method to a
particular example, which is a prolate spheroid.

**References:**

[1] M. Sieber and F. Steiner, Classical and quantum mechanics of a strongly
chaotic billiard system, Physica D, vol.44, pp. 248-266, August 1990.

[2] M. V. Berry, Quantizing a classically ergodic system: Sinai-s billiard and
the KKR method, Ann.Phys., vol. 131, pp. 163-216, 1981.

[3] D. A. McGrew and W. Bauer, Constraint operator solution to quantum
billiard problem, Phys. Rev. E, vol. 54, pp. 5809-5818, November 1996.

[4] M. Sieber and F. Steiner, Quantum chaos in the hyperbola billiard, Phys.
Lett. A, vol. 148, pp. 415-420, September 1990.

[5] M. Robnik, Quantizing a generic family of billiards with analytic boundaries,
J. Phys. A: Math. Gen., vol. 17, pp. 1049-1074, 1984.

[6] T. Papenbrock, Numerical study of a three-dimensional generalized
stadium billiard, Phys. Rev. E, vol. 61, pp. 4626-4628, April 2000.

[7] H. Primak and U. Smilansky, The quantum three-dimensional Sinai
billiard - A semiclassical analysis, Phys. Rep., vol. 327, pp. 1-107,
April 2000.

[8] W. A. Strauss, Partial Differential Equations, New York:
John-Wiley, 1992.

[9] H. Tas┬©eli, İInci M. Erhan and O┬¿ . Ug╦ÿur, An eigenfunction expansion for
the Schr┬¿odinger equation with arbitrary non-central potentials, J. Math.
Chem., vol. 32, pp. 323-338, November 2002.

[10] M. Abramovitz and I. A. Stegun , Handbook of Mathematical Functions,
New York:Dover, 1970.

[11] S. A. Moszkowski, Particle states in spheroidal nuclei, Phys Rev. vol.
99, pp. 803-809, 1955.