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Maclaurin spectral results on rank one symmetric spaces of noncompact type. (English) Zbl 1507.58012

Summary: This paper presents new spectral properties of the Laplacian on \(N\)-dimensional \((N\geq 1)\) noncompact rank one symmetric spaces \(\mathscr{X}\). Two new classes of spectral functions associated with the heat kernel \(K_N(t,\varrho)\) (\(t>0,\,0\leq \varrho <\infty\)) on \(\mathscr{X}\) are developed, namely, the Maclaurin heat coefficients \(\mathsf{b}_{2\ell}^N=\mathsf{b}_{2\ell}^N(t)\) (\(t>0,\,\ell \geq 1\)) and the associated zeta functions (Maclaurin zeta functions) \(\zeta_{2\ell}^N=\zeta_{2\ell}^N(z)\) \((z\in \mathbb{C},\,\ell \geq 1\)). The Maclaurin coefficient (which is the coefficient appearing in the Maclaurin expansion of the heat kernel) is expressed via Jacobi coefficients as a finite sum of higher order derivatives of the classical partition function \(\mathsf{K}_N(t):=K_N(t,0)\). Relation of this Maclaurin heat coefficient to the Minakshisundaram-Pleijel heat coefficient is given. We further express the Maclaurin heat coefficient, and by extension, the heat kernel, in terms of a class of spectral functions containing spectral information about the Minakshisundaram-Pleijel heat coefficient. On the other hand, the Maclaurin zeta function, which is obtained as the Mellin transform of the Maclaurin heat coefficient \(\mathsf{b}_{2\ell}^N(t)\), is asymptotically expressed near the origin, in terms of a class of polynomials and power series encoding structural properties of the classical zeta function on \(\mathscr{X}\). The first few of these polynomials and power series are computed explicitly.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
35A08 Fundamental solutions to PDEs
35C05 Solutions to PDEs in closed form
35C10 Series solutions to PDEs
35C15 Integral representations of solutions to PDEs
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
35K08 Heat kernel
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