Energy levels of a quantal particle enclosed in N identical, mirror symmetric wells of a periodic potential. Numerical test of phase-integral formulas

An interesting structure prevails for the energy levels of a quantal partic le in a periodic potential with N (greater than or equal to 2) mirror symme tric wells separated by N-1 mirror symmetric barriers, when the logarithmic derivative of the wave function is given at corresponding (periodically an d mirror symmetrically situated) points in the barrier to the left of the f irst well and in the barrier to the right of the Nth well. It is shown that the quantization conditions that one obtains for these energy levels by me ans of a careful and rigorous phase-integral treatment are capable of givin g extremely accurate results. The accuracy obtainable is demonstrated for N =3 by comparison with numerically exact results, which were obtained by mea ns of the extended version of the phase-amplitude method presented in an Ap pendix. In the concluding section we summarize the results and point out un expected features of the energy spectrum and the wave functions. Two differ ent boundary conditions, commonly used in the theory of crystals, and close ly related to the present investigation, are also discussed there. (C) 1999 American Institute of Physics. [S0022-2488(99)02303-8].