THE POLYGON QUANTUM-BILLIARD PROBLEM

The present work addresses the quantum polygon billiard problem with a ttention given to analytic and degeneracy properties of energy eigenst ates. The ''polygon ground-state theorem'' is proven which states that the only polygons that contain respective ground states that are anal ytic in the closed domain of the entire polygon are the ''elemental po lygons'' (defined in the text). The ''polygon first excited-state theo rem'' is established which states that for every N-sided regular polyg on, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to es tablish the second component of this theorem which indicates that the space of first excited states for the N-sided regular polygon is spann ed by any two of these N nodal-diagonal eigenstates (i.e., the first e xcited state is twofold degenerate). At various levels of the discussi on attention is drawn to the correspondence between the quantum and cl assical solutions for these configurations. Thus, for example, corresp ondence is demonstrated in the common source of integrability in the c lassical and quantum domains for three inclusive, nonoverlapping class es of polygons. Stemming from the preceding conclusions, a discussion is included on the possibility of employing transverse magnetic (TM) m odes in a metal wave-guide to determine if an equivalent quantum billi ard configuration is chaotic.