TIME-DEPENDENT HARMONIC-OSCILLATOR CONFINED IN A BOX

In formulating the quantum-mechanical analog of the Fermi accelerator, one encounters the problem of determination of the eigenvalues of a harmonic oscillator confined to a finite region of space with time-dependent real or imaginary frequency. Here the invariants found by Lewis [J. Math. Phys. 9, 1976 (1968)] and by Lewis and Riesenfeld [J. Math. Phys. 10, 1403 (1969)] are also constants of motion; however, they are not very useful in obtaining the eigenvalues. In their place the Heisenberg equations of motion are used to describe the time evolution of this system and show how the spacing between neighboring eigenvalues changes as a function of time.