Abstract
The goal of this chapter is to prove the Spectral Theorem (Theorem 1.6).
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Notes
- 1.
A continuous operator (T, D T ) on D T naturally extends to a bounded operator of H to itself.
- 2.
The usual Euclidean Laplacian is negative definite. Throughout this text the hyperbolic Laplacian that we consider is the geometric Laplacian, which is positive definite. This accounts for the difference in sign here.
- 3.
The two interior parentheses on the right-hand side of (3.5) are the “normal derivatives” which arise in the general Green’s formula on a Riemannian manifold.
- 4.
Proposition 3.2 implies in particular that the functions F s and F 1−s are equal for s ∈ [0, 1]; we shall in general assume that \(s\leqslant 1/2\).
- 5.
Recall that ρ(z, w) is the hyperbolic distance between two points z and w in \(\mathcal{H}\).
- 6.
Here the term “invariant” comes from the fact that for every g ∈ G and for all \(z,w \in \mathcal{H}\) we have k(gz, gw) = k(z, w).
- 7.
One could just as well refer to the proof of Lemma 3.29.
- 8.
This can be easily seen if one recalls that this particular hyperbolic circle is a Euclidean circle whose maximal ordinate is on the imaginary axis.
- 9.
Note the change in convention relative to the introduction.
- 10.
The notation “PW” is in reference to the classical Paley-Wiener theorem, see [106].
- 11.
One can just as well verify this directly by using the formula for \(p_{\mathcal{H}}(z,w,t)\), but it is a bit tedious.
- 12.
This is a fundamental property of hyperbolic geometry.
- 13.
Recall that | ⋅ | denotes the Hilbert norm of \(L^{2}(\varGamma \setminus \mathcal{H})\).
- 14.
Note that \(\varphi _{0}\) is the function constantly equal to \(1/\sqrt{\mathop{\mathrm{area } }\nolimits (S)}\).
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Bergeron, N. (2016). Spectral Decomposition. In: The Spectrum of Hyperbolic Surfaces. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27666-3_3
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