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I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of Borthwick's book, where one finds the statement that (roughly speaking) infinite area hyperbolic surfaces with finitely many cusps and funnels have finitely many eigenvalues less than $1/4$, and the continuous spectrum is $[1/4, \infty)$ with no embedded eigenvalues.

My question is, what is the corresponding statement for higher dimensional infinite volume (let's say geometrically finite) hyperbolic manifolds? Borthwick mentions that it is contained in a collection of papers of Lax and Phillips, but I am not at all familiar with this work, and I was hoping for a more precise reference. Thanks!

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In dimension $n$, there are at most finitely many eigenvalues in $[ 0, (n-1)^2/4 )$ and that the continuous spectrum is $[ (n-1)^2/4 , \infty )$ with no embedded eigenvalues. The following survey article has a good discussion and an extensive bibliography (including the papers of Lax and Phillips that you asked for):

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    $\begingroup$ Excellent, exactly what I wanted. For anyone else interested, it is Theorem 3.1, page 7 of Perry's article. $\endgroup$
    – SMS
    Commented May 9, 2022 at 18:52

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