Growth of heat trace coefficients for locally symmetric spaces

P. GILKEY; R. J. MIATELLO

JOURNAL OF MATHEMATICAL PHYSICS

AMER INST PHYSICS

Lugar: New York; Año: 2012 vol. 53 p. 1 - 11

0022-2488

Abstract. We study the asymptotic behavior of the heat trace coecients an as n ! 1 for the scalar Laplacian in the context of locally symmetric spaces. We show that if a locally symmetric space is modeled on a noncompact type symmetric space X = G=K such that G has one conjugacy class of Cartan subgroups then these coecients are O( Cn n! ) for some C > 0. In contrast, we prove that for even dimensional locally rank 1-symmetric spaces one has janj  C0n
n! for some C0 > 0. We conjecture this is the growth for general noncompact type symmetric spaces unless all Cartan subgroups in G are conjugate. These spaces also have the distinguishing property that the Huygens principle for the shifted wave equation holds (n odd). These examples show that growth estimates conjectured by Berry and Howls [1] are sharp. We also construct examples of locally symmetric spaces which are not irreducible, which are not at, and so that only a nite number of the an are non-zero. MSC: 58J50, 58J35