TY - JOUR

T1 - Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori

AU - T. Sardari, Naser

N1 - Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Let Q(X) be any integral primitive positive definite quadratic form in k variables, where k≥ 4 , and discriminant D. For any integer n, we give an upper bound on the number of integral solutions of Q(X) = n in terms of n, k, and D. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus Td for d≥ 5. This conjecture is motivated by the work of Berry [2,3] on the semiclassical eigenfunction hypothesis.

AB - Let Q(X) be any integral primitive positive definite quadratic form in k variables, where k≥ 4 , and discriminant D. For any integer n, we give an upper bound on the number of integral solutions of Q(X) = n in terms of n, k, and D. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus Td for d≥ 5. This conjecture is motivated by the work of Berry [2,3] on the semiclassical eigenfunction hypothesis.

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U2 - 10.1007/s00220-017-3044-1

DO - 10.1007/s00220-017-3044-1

M3 - Article

AN - SCOPUS:85035145746

SN - 0010-3616

VL - 358

SP - 895

EP - 917

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -