Dennis A. Hejhal's Books

This monograph presents many interesting results, old and new, about theta functions, Abelian integrals and kernel functions on closed Riemann surfaces. It begins with a review of classical kernel function theory for plane domains. Next there is a discussion of function theory on closed Riemann surfaces, leading to explicit formulas for Szegö kernels in terms of the Klein prime function and theta functions. Later sections develop explicit relations between the classical Szegö and Bergman kernels and between the Szegö and modified (semi-exact) Bergman kernels. The author's results allow him to solve an open problem mentioned by L. Sario and K. Oikawa in 1969.

This paper is concerned with the spectral theory of the Laplacian as the underlying Riemann surface is "pinched down" to a surface with nodes. The problem is attacked from the (general) standpoint of regular b-groups and the Selberg trace formula.

Paper I is concerned with computational aspects of the Selberg trace formalism, considering the usual type of eigenfunction and including an analysis of pseudo cusp forms and their residual effects. Paper II examines the modular group PSL (2, [bold]Z), as such groups have both a discrete and continuous spectrum. This paper only examines the discrete side of the spectrum.

Book's by Dennis A. Hejhal

Theta Functions, Kernel Functions and Abelian Integrals

Theta Functions, Kernel Functions and Abelian Integrals

Regular $b$-Groups, Degenerating Riemann Surfaces, and Spectral Theory

Regular $b$-Groups, Degenerating Riemann Surfaces, and Spectral Theory

Eigenvalues of the Laplacian for Hecke Triangle Groups

Eigenvalues of the Laplacian for Hecke Triangle Groups

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