The local geometric asymptotics of continuum eigenfunction expansions. III. Boundary effects and coefficient singularities
S. A. Fulling · Journal of Physics A Mathematical and General · 1983
For pt.II see SIAM J. Math. Anal., vol.14, no.4, p.780-95 (1983). The effect of a boundary condition on the spectral density of a differential operator in one dimension is computed directly from the asymptotic behaviour of the eigenfunctions. From the properly normalised eigenfunction expansion, the contribution of the boundary to the diagonal value of the heat kernel at a point is obtained, and some properties of the special functions arising thereby are derived. A discontinuity, or other singularity, in the coefficient function of the operator is shown to have spectral effects quite analogous to those of a boundary, and the additional effects due to the coexistence of a boundary and a discontinuity are investigated to lowest order.