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](https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=328585&sort=Newest&vfpref=html&r=6&mx-pid=3497464), 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!



Spectral theory of infinite volume hyperbolic manifolds
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):


* Perry, Peter: [The spectral geometry of geometrically finite
hyperbolic manifolds](http://dx.doi.org/10.1090/pspum/076.1/2310208). (English summary) Spectral theory and
mathematical physics: a Festschrift in honor of Barry Simon's 60th
birthday, 289–327, Proc. Sympos. Pure Math., 76, Part 1, Amer. Math.
Soc., Providence, RI, 2007. [MR2310208](https://mathscinet.ams.org/mathscinet-getitem?mr=2310208)


