Recent Developments in Quantum Mechanics pp 265-277 | Cite as

# Eigenfunction Expansions for Hyperbolic Laplacians

## Abstract

The geometry near infinity of a non-compact complete Riemannian manifold is reflected in the essential spectrum its Laplace operator. The nature of this correspondence, however, is subtle, and the manifolds whose Laplace operators have a well understood spectral theory are ones whose structure near infinity is simple. In this lecture I would like to explain some recent work of P. Hislop, P. Perry and myself on the Laplacian for three-dimensional non-compact hyperbolic manifolds. These manifolds are are simple near infinty, in the sense of being geometrically finite, but somewhat less simple than the ones that had been considered previously, in that they have cusps of non-maximal rank.

## Keywords

Laplace Operator Eisenstein Series Hyperbolic Manifold Kleinian Group Eigenfunction Expansion## Preview

Unable to display preview. Download preview PDF.

## References

- [Al]S. Agmon,
*On the spectral theory of the Laplacian on non-compact hyperbolic manifolds*, Journées `Equations aux derivées partielles’ (Saint-Jean de Monts, 1987 ), Exp. No. XVII. École Polytechnique, Palaiseau, 1987.Google Scholar - [A2]S. Agmon,
*Spectral Theory of Schrödinger Operators on Euclidean and Non-Euclidean Spaces*C.P.A.M.**39**(1986), Number S, Supplement.Google Scholar - [CFKS]H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon,
*Schrödinger Operators, with Application to Quantum Mechanics and Global Geometry*, Springer Texts and Monographs in Physics, Springer-Verlag, New York, 1986.Google Scholar - [F]L. Faddeev,
*Expansion in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Labacevskii plane*, Trudy Moscow. Mat. Obshch.**17**, (1967), 323–350.MathSciNetzbMATHGoogle Scholar - FHP1] R. Froese, P. Ilislop, P. Perry, A
*Mourre Estimate and Related Bounds for Hyperbolic Manifolds with Cusps of Non-maximal Rank*,to appear in J. Funct. Anal.Google Scholar - [FHP2]R. Froese, P. Hislop, P. Perry,
*The Laplace Operator on a Hyperbolic Three Manifold with Cusps of Non-Maximal Rank*,preprint.Google Scholar - [LP1]P. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions, Annals of Math Studies 87, Princeton University Press, 1976.Google Scholar
- [LP2]P. Lax. and R.S. Phillips,
*Translation representation for automorphic solutions of the non-Euclidean wave equation, I, II and III*, Comm. Pure Appl. Math.**37**, 303–328, 1984MathSciNetzbMATHCrossRefGoogle Scholar - [LP3]P. Lax. and R.S. Phillips,
*Translation representation for automorphic solutions of the non-Euclidean wave equation, I, II and III*, Comm. Pure Appl. Math.**37**, 779–813, 1984MathSciNetzbMATHCrossRefGoogle Scholar - [LP4]P. Lax. and R.S. Phillips,
*Translation representation for automorphic solutions of the non-Euclidean wave equation*,*I, II and III*, Comm. Pure Appl. Math.**38**, 179–208, 1985.MathSciNetzbMATHCrossRefGoogle Scholar - [LP5]P. Lax and R.S. Phillips,
*Translation representation for automorphic solutions of the non-Euclidean wave equation*,*IV*,PreprintGoogle Scholar - [M]H. Maass,
*Ober eine neue Art von nichtanalytischen automorphen Functionen und die Bestimmumng Dirichletscher Reihen durch Functionalgleichungen*, Math. Ann.**121**, (1949), 141–183.MathSciNetzbMATHCrossRefGoogle Scholar - [Mal]N. Mandouvalos,
*The Theory of Eisenstein Series for Kleinian Groups*, In*The Selberg Trace Formula and Related Topics*, Hejhal, Sarnak and Terras eds., (Contemporary Mathematics**53**, 1986 ) 357–370.CrossRefGoogle Scholar - [Ma2]N. Mandouvalos,
*Scattering operator, inner product formula, and “Maass-Selberg” relations for Kleinian groups*, AMS Memoir**400**, 1989.Google Scholar - [Ma3]N. Mandouvalos,
*Spectral theory and Eisenstein series for Kleinian groups*, Cambridge Unversity preprint, 1986.Google Scholar - [MM]R. Mazzeo and R. Melrose,
*Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature*,J. Funct. Anal.**75**, 260–310,1987.MathSciNetCrossRefGoogle Scholar - [P]S.J. Patterson,
*The Laplacian Operator on a Riemann Surface*, I, II and III, Compositio Math. 31 (1975), 83–107; 32 (1976), 71–112, 33 (1976), 227–259.Google Scholar - [P]P. Perry,
*The Laplace operator on a hyperbolic manifold, II. Eisenstein series and the scattering matrix*, J. Reine Angew. Math. 398, 67–91, 1989.MathSciNetzbMATHGoogle Scholar - W. Roelcke,
*Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene*, Teil I, Math. Ann. 167 (1966), 292–337: Teil II, Math. Ann. 168, (1967), 261–324.Google Scholar - [T]A. Terras,
*Harmonic Analysis on Symmetric Spaces and Applications I*, Springer-Verlag, 1985.zbMATHCrossRefGoogle Scholar