Spectral Zeta Function Research Articles

An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics

We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of δ-like potentials with strength κ > 0 on the half line and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval In = [0, ln], , to each edge with length . We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each κ > 0 and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by λn = n2, , with multiplicities d(n), where d(n) denotes the divisor function. We can thus relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics for the eigenvalue counting function. Based on an exact trace formula, the Voronoï summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true.

Small mass expansion of functional determinants on the generalized cone

In this paper we compute the small mass expansion for the functional determinant of a scalar Laplacian defined on the bounded, generalized cone. In the framework of zeta function regularization, we obtain an expression for the functional determinant valid in any dimension for both Dirichlet and Robin boundary conditions in terms of the spectral zeta function of the base manifold. Moreover, as a particular case, we specify the base to be a d-dimensional sphere and present explicit results for d = 2, 3, 4, 5.

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called thecomplexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study ofI-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

Spectral functions for the Schrödinger operator on R+ with a singular potential

In this article we analyze the spectral zeta function, the heat kernel, and the resolvent of the operator −d2/dr2+κ/r2+r2 over the interval (0,∞) for κ≥−1/4. Depending on the self-adjoint extension chosen, nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed. In particular, for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s=−k, k∊N0, do occur. This implies that the small-t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk/(ln t)ℓ+1 for k,ℓ∊N0. The corresponding statements for the resolvent are also shown. Furthermore, we evaluate the zeta determinant of the operator for all values of κ and any self-adjoint extension.

Spectral zeta function of a sub-Laplacian on product sub-Riemannian manifolds and zeta-regularized determinant

We analyze the spectral zeta function for sub-Laplace operators on product manifolds M × N . Starting from suitable conditions on the zeta functions on each factor, the existence of a meromorphic extension to the complex plane and real analyticity in a zero neighbourhood is proved. In the special case of N = S 1 and using the Poisson summation formula, we obtain expressions for the zeta-regularized determinant. Moreover, we can calculate limit cases of such determinants by inserting a parameter into our formulas. This is a generalization of results in Furutani and de Gosson (2003) [1] and in particular it applies to an intrinsic sub-Laplacian on U ( 2 ) ≅ S 3 × S 1 induced by a sum of squares of canonical vector fields on S 3 ; cf. Bauer and Furutani (2008) [2]. Finally, the spectral zeta function of a sub-Laplace operator on Heisenberg manifolds is calculated by using an explicit expression of the heat kernel for the corresponding sub-Laplace operator on the Heisenberg group; cf. Beals et al. (2000) [18] and Hulanicki (1976) [19].

Trace of heat kernel, spectral zeta function and isospectral problem for sub-laplacians

In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ℂn+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.

A variation of multiple $L$-values arising from the spectral zeta function of the non-commutative harmonic oscillator

A variation of multiple L-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function can be written in terms of the gamma functions. This result enables us to obtain explicit evaluations of them.

A Negative Mass Theorem for Surfaces of Positive Genus

Let M be a closed surface. For a metric g on M, denote the Laplace-Beltrami operator by Δ = Δg. We define trace \({\Delta^{-1} = \int_M m(p)dA}\) , where dA is the area element for g and m(p) is the Robin constant at the point \({p \in M}\) , that is the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off. Since trace Δ−1 can also be obtained by regularization of the spectral zeta function, it is a spectral invariant. Heuristically it represents the sum of squares of the wavelengths of the surface. We define the Δ-mass of (M, g) to equal \({({\rm trace} \Delta_g^{-1} - {\rm trace} \Delta_{S^{2},A}^{-1})/A}\) , where \({\Delta_{S^2,A}}\) is the Laplacian on the round sphere of area A. This is an analog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus, the minimum of the Δ-mass on each conformal class is negative and attained by a smooth metric. For this minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.

Identities for Sums of a q-Analogue of Polylogarithm Functions

A q-analogue of the polylogarithm function is introduced via a consideration of the spectral zeta-function of the quantum group SU q (2). We derive certain identities for linear and non-linear combinations of the q-analogue of polylogarithm functions with negative exponents.

Quantum mass correction for the twisted kink

We present an analytic result for the 1-loop quantum mass correction in semiclassical quantization for the twisted ϕ4-kink on S1 without explicit knowledge of the fluctuation spectrum. For this purpose we use the contour integral representation of the spectral zeta function. By solving the Bethe ansatz equations for the n = 2 Lamé equation, we obtain an analytic expression for the corresponding spectral discriminant. We discuss the renormalization issues of this model. An energetically preferred size for the compact space is finally obtained.

A Negative Mass Theorem for the 2-Torus

Let M be a closed surface. For a metric g on M, denote the area element by dA and the Laplace-Beltrami operator by Δ = Δ g . We define the Robin mass m(p) at the point $${{p \in\, M}}$$ to be the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off, and we define trace $${{\Delta^{-1}=\int_M m(p)\, dA}}$$ . This regularized trace can also be obtained by regularization of the spectral zeta function and is hence a spectral invariant which heuristically measures the total wavelength of the surface.We define the Δ-mass of (M, g) to equal $${{({\rm trace} \, \Delta_{g}^{-1}-{\rm trace} \, \Delta_{S^2,A}^{-1})/A}}$$ , where $${{\Delta_{S^2,A}}}$$ is the Laplacian on the round sphere of area A. This scale invariant quantity is a non-trivial analog for closed surfaces of the ADM mass for higher dimensional asymptotically flat manifolds.In this paper we show that in each conformal class $${{\mathcal C}}$$ for the 2-torus, there exists a metric with negative Δ-mass. From this it follows that the minimum of the Δ-mass on $${{\mathcal C}}$$ is negative and attained by some metric $${{g\in\mathcal C}}$$ . For this minimizing metric g, one gets a sharp logarithmic Hardy-Littlewood-Sobolev inequality and an Onofri-type inequality.We remark that if the flat metric in $${{\mathcal C}}$$ is sufficiently long and thin then the minimizing metric g is non-flat. The proof of our result depends on analyzing the ordinary differential equation $${{\phi''=1-e^\phi}}$$ which is equivalent to h′′ = 1 − 1/h. The solutions are periodic and we need to establish quite delicate, asymptotically sharp inequalities relating the period to the maximum value.

Spectral analysis and geometry of a sub-Riemannian structure on [formula omitted] and [formula omitted

The purpose of this paper is to study the spectral properties of a sub-Laplacian on S 3 , i.e., we discuss the analytic continuation of its spectral zeta function, give explicit expressions of the residues and especially, we provide an expression of the zeta-regularized determinant of the sub-Laplacian on S 3 . Also, we describe sub-Riemannian curves on S 3 based on the Hopf bundle structure, together with a proof of Chow’s theorem for this case in a strong sense (= connecting property by globally smooth curves). A characterization of sub-Riemannian geodesics on S 3 via an isoperimetric problem through the Hopf bundle is explained. Incidentally, we introduce a hypo-elliptic operator on P 1 C descended from the sub-Laplacian on S 3 , which we call a spherical Grushin operator. We determine the subspace where it degenerates and give an expression of the trace of its heat kernel by making use of the trace of the heat kernel of the sub-Laplacian. In case of S 7 , we limit ourselves to present the spectral zeta function of a sub-Laplacian.

A special value of the spectral zeta function of the non-commutative harmonic oscillators

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.

Vibration modes of 3n-gaskets and other fractals

We rigorously study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals, which include the Sierpinski gasket and other 3n-gaskets. We consider the classical Laplacian on fractals which generalizes the usual one-dimensional second derivative, is the generator of the self-similar diffusion process, and has possible applications as the quantum Hamiltonian. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations. Our analysis, in particular, allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues (i.e., integrated density of states). We consider such examples as the level-3 Sierpinski gasket, a fractal 3-tree, and the diamond fractal.

Zeta regularized determinant of the Laplacian for classes of spherical space forms

We derive the spectral zeta function in terms of certain Dirichlet series for a variety of spherical space forms M G . Extending results in [C. Nash, D. O’Connor, Determinants of Laplacians on lens spaces, J. Math. Phys. 36 (3) (1995) 1462–1505] the zeta-regularized determinant of the Laplacian on M G is obtained explicitly from these formulas. In particular, our method applies to manifolds of dimension higher than 3 and it includes the case where G arises from the dihedral group of order 2 m . As a crucial ingredient in our analysis we determine the dimension of eigenspaces of the Laplacian in form of some combinatorial quantities for various infinite classes of manifolds from the explicit form of the generating function in [A. Ikeda, On the spectrum of a Riemannian manifold of positive constant curvature, Osaka J. Math. 17 (1980) 75–93].

On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator

The spectral zeta function for the so-called noncommutative harmonic oscillator is able to be meromorphically extended to the whole complex plane, having only one simple pole at the same point s = 1 where Riemann's zeta function ζ( s) has, and possesses a trivial zero at each nonpositive even integer. The essential part of its proof is sketched. A new result is also given on the lower and upper bounds of the eigenvalues of the noncommutative harmonic oscillator.

Mini-Workshop: Zeta Functions, Index and Twisted K-Theory; Interactions with Physics

This mini-workshop brought together number theorists, analysts, geometers and mathematical physicists to discuss current issues at the common boundary of mathematics and physics. Topics covered included the number theoretic and algebraic structures underlying renormalization, twisted K-theory and higher algebraic structures, modular forms, and arithmetic and spectral zeta functions. A particular theme was around developing interconnections between arithmetic (multiple) zeta functions, spectral zeta functions associated with elliptic operators (and related spectral invariants such as spectral flow) and current issues in physics such as renormalization and mirror symmetry. Multiple zeta functions appear in index theory and K -theory via their relation to anomalies, in number theory in their relation to polylogarithms, in renormalization questions in perturbative quantum field theory and Hopf algebras, in duality issues and in twisted K -theory for index theorems for projective families of elliptic operators, thereby providing a rich set of overlapping topics with common analytical issues. This meeting was organized around one hour talks, four each day, with plenty of time between talks for informal discussion and a 45 minute talk in the afternoon for students; three graduate students were among the 16 participants. Some participants lectured for two hours in order to have time to introduce the audience to the subject before entering the technical details. The organizers and participants would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing a pleasant and stimulating enviroment for this meeting.

Spectral zeta functions of fractals and the complex dynamics of polynomials

We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half plain with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta functions of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

Multifractal analysis of the spectral measure of the Thue–Morse sequence: a periodic orbit approach

The Fourier spectral density of the Thue?Morse sequence is reinterpreted as the invariant measure of a stochastic dynamical system. Based on this fact, its generalized (R?nyi) dimension and f(?) statistics are calculated with high precision by cycle expansions of spectral determinant and dynamical zeta function. ?q at integer values of q are also computed in an operator scheme and the asymptotic result in the large-q limit is derived.