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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2025 Volume 89, Issue 5, Pages 165–180 (Mi im9649)

Resonances and discrete spectrum of the Laplace operator on hyperbolic surfaces

D. A. Popov

Lomonosov Moscow State University, Belozersky Research Institute of Physico-Chemical Biology

Abstract: The spectrum of the Laplace operator on a non-compact hyperbolic Riemann surface of finite measure is studied. A sufficient condition for the discrete spectrum to be infinite is obtained. It is shown that this condition holds near the point $\Gamma_0(N)/H$, $N=p_1\cdots p_r$, of the Teichmüller space.

Keywords: Laplace operator, spectrum, resonance, cofinite group, Roelcke's conjecture.

UDC: 517.484.5

MSC: 11F72

Received: 11.09.2024
Revised: 27.12.2024

DOI: 10.4213/im9649


 English version:
Izvestiya: Mathematics, 2025, 89:5, 1024–1039

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© Steklov Math. Inst. of RAS, 2026