ASYMPTOTICS OF THE EIGENVALUES OF THE LAPLACIAN AND QUASIMODES. A SERIES OF QUASIMODES CORRESPONDING TO A SYSTEM OF CAUSTICS CLOSE TO THE BOUNDARY OF THE DOMAIN
V. F. Lazutkin · Mathematics of the USSR-Izvestiya · 1973
For a bounded convex domain in the plane, asymptotic formulas with error tending to zero are constructed for a certain series of eigenvalues of the Laplacian with zero boundary conditions. The boundary of the domain is assumed to be sufficiently smooth. It is proved that 0,$ SRC=http://ej.iop.org/images/0025-5726/7/2/A12/tex_im_1949_img1.gif/>where is the number of eigenvalues (with multiplicities taken into account) less than and is the number of those eigenvalues for which an asymptotic expansion has been found.