Perturbation theory for the Laplacian on automorphic functions (1992)

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma subset-of PSL left-parenthesis 2 comma double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>PSL(2, </mml:mtext> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>)</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \subset {\text {PSL(2, }}\mathbb {R}{\text {)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a discrete subgroup with quotient <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma minus upper H"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \backslash H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite volume but not compact. The spectrum of the Laplacian on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> automorphic functions is unstable under perturbations; however, it becomes much more manageable when the scattering frequencies are adjoined (with multiplicity equal to the order of the pole of the determinant of the scattering matrix at these points). This augmented set shows up in a natural way in a one-sided version of the Selberg trace formula and is the actual spectrum of the generator of a cut-off wave equation. Applying standard perturbation theory to this operator, it is proved that the augmented spectrum is real analytic in Teichmüller space. The same operator is used to derive Fermi’s Golden Rule in this setting. It turns out that the proper multiplicity to be attached to the Laplacian eigenvalue at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="one fourth"> <mml:semantics> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {1}{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is twice the dimension of cusp forms plus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu equals trace left-bracket normal upper Phi plus upper I right-bracket slash 2"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> tr</mml:mtext> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> + </mml:mtext> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>/2</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu = {\text { tr}}[\Phi {\text { + }}I]{\text {/2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the scattering matrix at this point. It is shown that the generic value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Teichmüller space of the once punctured torus and the six-times punctured sphere is zero. This is also true of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi>χ</mml:mi> <mml:annotation encoding="application/x-tex">\chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-twisted spectral problem, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi>χ</mml:mi> <mml:annotation encoding="application/x-tex">\chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a character for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.