Geodesic
Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces
Canadian journal of mathematics, Tome 45 (1993) no. 4, pp. 709-726

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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.
DOI : 10.4153/CJM-1993-040-0
Mots-clés : 58G25, 60J80, 30B70, 39A10 
Edward, Julian. Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces. Canadian journal of mathematics, Tome 45 (1993) no. 4, pp. 709-726. doi: 10.4153/CJM-1993-040-0
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