##
**Selberg trace formulae and equidistribution theorems for closed geodesics and Laplace eigenfunctions: Finite area surfaces.**
*(English)*
Zbl 0753.11023

Mem. Am. Math. Soc. 465, 102 p. (1992).

Let \(\mathbb{H}\) be the upper half-plane and \(\Gamma<G:=\text{PSL}_ 2(\mathbb{R})\) a (non-cocompact) cofinite discrete group. The work under review is an extension of a previous article by the author [Duke Math. J. 59, 27-81 (1989; Zbl 0686.10024)] in which the case of a cocompact group \(\Gamma\) was discussed.

An important problem considered here is to determine the precise asymptotic distribtion of closed geodesics in the unit tangent bundle. The author’s aim is a version of Bowen’s equidistribution theory of closed geodesics. The problem is to analyze the growth of \(\psi(\sigma,T)=\sum_{L(\gamma)\leq T}\int_ \gamma\sigma\) where \(\sigma\) is an automorphic form on \(\Gamma\backslash G\) and where the sum extends over all closed geodesics \(\gamma\) of length \(L(\gamma)\) less than \(T\). Using his generalization of the Selberg trace formula the author proves an asymptotic expansion for \(\psi(\sigma,T)\) with remainder term. The leading terms of this expansion are determined by the exceptional eigenvalues \(\lambda_ k<{1\over 4}\) and eigenfunctions \(u_ k\) \((k=1,\ldots,M)\) of \(-\Delta\) on \(L^ 2(\Gamma\backslash\mathbb{H})\). In particular, there appear certain matrix coefficients \(\langle Op(\sigma)u_ k,u_ k\rangle\) of a certain pseudodifferential operator \(Op(\sigma)\) associated with \(\sigma\). For \(\sigma=1\) the author’s result agrees with the prime geodesic theorem, and if \(\sigma\perp 1\) the author’s result yields a new proof of the fact that closed geodesics are uniformly distributed.

A closely related problem is to determine the distribution of eigenfunctions (in a microlocal sense). A main result is a version of the Lindelöf hypothesis for zeta functions of Rankin-Selberg type. A key rôle is played by certain analogues of Weyl’s law for the sums \[ N_ \Gamma(\sigma,T)=\sum_{| r_ j|\leq T}\langle Op(\sigma)u_ j,u_ j\rangle, \] where \(\lambda_ j={1\over 4}+r^ 2_ j\) runs through the eigenvalues of \(-\Delta\) and their continuous analogue \[ M_ \Gamma(\sigma,T)=-{1\over 4\pi}\int^ T_{-T}\langle Op(\sigma)E(\cdot\;,{1\over 2}+ir), E(\cdot\;,{1\over 2}+ir)\rangle\;dr. \] If \(\sigma\) is an Eisenstein series, these functions are defined via Zagier’s method of renormalizing products. The author’s analogues of Weyl’s law state that \[ (M_ \Gamma+N_ \Gamma)(\sigma,T)=O(T/\log T) \qquad \text{if } \sigma \text{ is cuspidal}, \]

\[ (M_ \Gamma+N_ \Gamma)(E(\cdot,s),T)=O(T^{3/2}) \qquad \text{ for Re} s={1\over 2}. \] In view of the technical complexity of the work (more than 8 pages of index of notations!) it is impossible to give more details here. The reader is advised to read the author’s careful introduction. (Several references in the text do not appear in the list of references).

An important problem considered here is to determine the precise asymptotic distribtion of closed geodesics in the unit tangent bundle. The author’s aim is a version of Bowen’s equidistribution theory of closed geodesics. The problem is to analyze the growth of \(\psi(\sigma,T)=\sum_{L(\gamma)\leq T}\int_ \gamma\sigma\) where \(\sigma\) is an automorphic form on \(\Gamma\backslash G\) and where the sum extends over all closed geodesics \(\gamma\) of length \(L(\gamma)\) less than \(T\). Using his generalization of the Selberg trace formula the author proves an asymptotic expansion for \(\psi(\sigma,T)\) with remainder term. The leading terms of this expansion are determined by the exceptional eigenvalues \(\lambda_ k<{1\over 4}\) and eigenfunctions \(u_ k\) \((k=1,\ldots,M)\) of \(-\Delta\) on \(L^ 2(\Gamma\backslash\mathbb{H})\). In particular, there appear certain matrix coefficients \(\langle Op(\sigma)u_ k,u_ k\rangle\) of a certain pseudodifferential operator \(Op(\sigma)\) associated with \(\sigma\). For \(\sigma=1\) the author’s result agrees with the prime geodesic theorem, and if \(\sigma\perp 1\) the author’s result yields a new proof of the fact that closed geodesics are uniformly distributed.

A closely related problem is to determine the distribution of eigenfunctions (in a microlocal sense). A main result is a version of the Lindelöf hypothesis for zeta functions of Rankin-Selberg type. A key rôle is played by certain analogues of Weyl’s law for the sums \[ N_ \Gamma(\sigma,T)=\sum_{| r_ j|\leq T}\langle Op(\sigma)u_ j,u_ j\rangle, \] where \(\lambda_ j={1\over 4}+r^ 2_ j\) runs through the eigenvalues of \(-\Delta\) and their continuous analogue \[ M_ \Gamma(\sigma,T)=-{1\over 4\pi}\int^ T_{-T}\langle Op(\sigma)E(\cdot\;,{1\over 2}+ir), E(\cdot\;,{1\over 2}+ir)\rangle\;dr. \] If \(\sigma\) is an Eisenstein series, these functions are defined via Zagier’s method of renormalizing products. The author’s analogues of Weyl’s law state that \[ (M_ \Gamma+N_ \Gamma)(\sigma,T)=O(T/\log T) \qquad \text{if } \sigma \text{ is cuspidal}, \]

\[ (M_ \Gamma+N_ \Gamma)(E(\cdot,s),T)=O(T^{3/2}) \qquad \text{ for Re} s={1\over 2}. \] In view of the technical complexity of the work (more than 8 pages of index of notations!) it is impossible to give more details here. The reader is advised to read the author’s careful introduction. (Several references in the text do not appear in the list of references).

Reviewer: J.Elstrodt (Münster)

### MSC:

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |