TY - JOUR
T1 - Quantum Bianchi-VII problem, Mathieu functions and arithmetic
AU - Veselov, A. P.
AU - Ye, Y.
N1 - Publisher Copyright:
© 2023 The Authors
PY - 2023/7
Y1 - 2023/7
N2 - The geodesic problem on the compact threefolds with the Riemannian metric of Bianchi-VII0 type is studied in both classical and quantum cases. We show that the problem is integrable and describe the eigenfunctions of the corresponding Laplace-Beltrami operators explicitly in terms of the Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. We use the results from number theory to discuss the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation with Bianchi-VI0 case (Sol-case in Thurston's classification) and with Bianchi-IX case, corresponding to the classical Euler top.
AB - The geodesic problem on the compact threefolds with the Riemannian metric of Bianchi-VII0 type is studied in both classical and quantum cases. We show that the problem is integrable and describe the eigenfunctions of the corresponding Laplace-Beltrami operators explicitly in terms of the Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. We use the results from number theory to discuss the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation with Bianchi-VI0 case (Sol-case in Thurston's classification) and with Bianchi-IX case, corresponding to the classical Euler top.
KW - Bianchi classification
KW - Classical and quantum integrability
KW - Geodesic flows
KW - Thurston's special geometries
UR - http://www.scopus.com/inward/record.url?scp=85152690766&partnerID=8YFLogxK
U2 - 10.1016/j.geomphys.2023.104830
DO - 10.1016/j.geomphys.2023.104830
M3 - Article
AN - SCOPUS:85152690766
SN - 0393-0440
VL - 189
JO - Journal of Geometry and Physics
JF - Journal of Geometry and Physics
M1 - 104830
ER -