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Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces

Published online by Cambridge University Press:  20 November 2018

Julian Edward
Julian Edward*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 1A1
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Abstract

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The Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. The spectrum of the Neumann operator is studied on the curves bounding a family of Riemann surfaces. The Neumann operator is shown to be isospectral to a direct sum of symmetric Jacobi matrices, each acting on l2(ℤ). The Jacobi matrices are shown to be isospectral to generators of bilateral, linear birth-death processes. Using the connection between Jacobi matrices and continued fractions, it is shown that the eigenvalues of the Neumann operator must solve a certain equation involving hypergeometric functions. Study of the equation yields uniform bounds on the eigenvalues and also the asymptotics of the eigenvalues as the curves degenerate into a wedge of circles.

Keywords

58G2560J8030B7039A10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 4 , 01 August 1993 , pp. 709 - 726
Copyright
Copyright © Canadian Mathematical Society 1993

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