Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces

Jean-Philippe Anker (1) , Hong-Wei Zhang (1)

1 IDP - Institut Denis Poisson

We estimate the bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank.

Keywords: $L^2$ spectrum; Poincaré series; Critical exponent; Heat kernel; Locally symmetric space

Domains: Mathematics [math]; Spectral Theory [math.SP]

Jean-Philippe Anker, Hong-Wei Zhang. Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces. 2020. ⟨hal-02865274v1⟩