Spectral Zeta Function Research Articles

Fractal zeta functions and complex dimensions of relative fractal drums

The theory of “zeta functions of fractal strings” has been initiated by the first author in the early 1990s and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called “distance zeta functions,” which since then has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of “tube zeta functions,” defined using the tube function of a fractal set. These three classes of zeta functions, under the name of “fractal zeta functions,” exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal.

Orthogonal polynomials associated with Coulomb wave functions

A new class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials have in the theory of Bessel functions. The orthogonality measure for this new class is described in detail. In addition, the orthogonality measure problem is discussed on a more general level. Apart from this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of certain formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) matrix JL whose eigenvalues coincide with the reciprocal values of the zeros of the regular Coulomb wave function FL(η,ρ). The spectral zeta function corresponding to the regular Coulomb wave function or, more precisely, to the respective tridiagonal matrix is studied as well.

Spherical designs and heights of Euclidean lattices

We study the connection between the theory of spherical designs and the question of extrema of the height function of lattices. More precisely, we show that a full-rank n-dimensional Euclidean lattice Λ, all layers of which hold a spherical 2-design, realises a stationary point for the height h(Λ), which is defined as the first derivative at the point 0 of the spectral zeta function of the associated flat torus ζ(Rn/Λ). Moreover, in order to find out the lattices for which this 2-design property holds, a strategy is described which makes use of theta functions with spherical coefficients, viewed as elements of some space of modular forms. Explicit computations in dimension n⩽7, performed with Pari/GP and Magma, are reported.

Quantum-induced interactions in the moduli space of degenerate BPS domain walls

In this paper quantum effects are investigated in a very special two-scalar field model having a moduli space of BPS topological defects. In a $(1+1)$-dimensional space-time the defects are classically degenerate in mass kinks, but in $(3+1)$ dimensions the kinks become BPS domain walls, all of them sharing the same surface tension at the classical level. The heat kernel/zeta function regularization method will be used to control the divergences induced by the quantum kink and domain wall fluctuations. A generalization of the Gilkey-DeWitt-Avramidi heat kernel expansion will be developed in order to accommodate the infrared divergences due to zero modes in the spectra of the second-order kink and domain wall fluctuation operators, which are respectively $N\times N$ matrix ordinary or partial differential operators. Use of these tools in the spectral zeta function associated with the Hessian operators paves the way to obtain general formulas for the one-loop kink mass and domain wall tension shifts in any $(1+1)$- or $(3+1)$-dimensional $N$-component scalar field theory model. Application of these formulae to the BPS kinks or domain walls of the $N=2$ model mentioned above reveals the breaking of the classical mass or surface tension degeneracy at the quantum level. Because the main parameter distinguishing each member in the BPS kink or domain wall moduli space is essentially the distance between the centers of two basic kinks or walls, the breaking of the degeneracy amounts to the surge in quantum-induced forces between the two constituent topological defects. The differences in surface tension induced by one-loop fluctuations of BPS walls give rise mainly to attractive forces between the constituent walls except if the two basic walls are very far apart. Repulsive forces between two close walls only arise if the coupling is approaches the critical value from below.

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.

Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

We investigate the existence of the meromorphic extension of the spectral zeta function of the Laplacian on self-similar fractals using the classical results of Kigami and Lapidus (based on the renewal theory) and new results of Hambly and Kajino based on the heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true in the examples that have been analyzed in the existing literature.

A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards

We show that the spectral zeta functions of inhomogeneous strings and drums can be calculated using Rayleigh-Schrödinger perturbation theory. The inhomogeneities that can be treated with this method are small but otherwise arbitrary and include the previously studied case of a piecewise constant density. In two dimensions the method can be used to derive the spectral zeta function of a domain obtained from the small deformation of a square. We also obtain exact sum rules that are valid for arbitrary densities and that correspond to the values taken by the spectral zeta function at integer positive values; we have tested numerically these sum rules in specific examples. We show that the Dirichlet or Neumann Casimir energies of an inhomogeneous string, evaluated to first order in perturbation theory, contain in some cases an irremovable divergence, but that the combination of the two is always free of divergences. Finally, our calculation of the Casimir energies of a string with piecewise constant density and of two perfectly conducting concentric cylinders, of similar radius, reproduce the results previously published.

An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an ε-neighborhood of a given number λ > 0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.

The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type I × f N

In this work we study the spectral zeta function associated with the Laplace operator acting on scalar functions defined on a warped product of manifolds of the type $I\times_{f} N$ where $I$ is an interval of the real line and $N$ is a compact, $d$-dimensional Riemannian manifold either with or without boundary. Starting from an integral representation of the spectral zeta function, we find its analytic continuation by exploiting the WKB asymptotic expansion of the eigenfunctions of the Laplace operator on $M$ for which a detailed analysis is presented. We apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion.

Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals

We investigate the spectral zeta function of fractal differential operators such as the Laplacian on the unbounded (i.e. infinite) Sierpinski gasket and a self-similar Sturm–Liouville operator associated with a fractal self-similar measure on the half-line. In the latter case, Sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables, called the renormalization map. We obtain a factorization of the spectral zeta function of such an operator, expressed in terms of the Dirac delta hyperfunction, a geometric zeta function, and the zeta function associated with the dynamics of the corresponding renormalization map, viewed either as a polynomial function on (in the first case) or (in the second case) as a polynomial on the complex projective plane, . Our first main result extends to the case of the fractal Laplacian on the unbounded Sierpinski gasket a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended by Teplyaev to the bounded (i.e. finite) Sierpinski gasket and some other decimable fractals. Furthermore, our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal Sturm–Liouville operators. Moreover, as a corollary, in the very special case when the underlying self-similar measure is the Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in , thereby extending to several variables an analogous result by Teplyaev.

Finite pseudo orbit expansions for spectral quantities of quantum graphs

We investigate spectral quantities of quantum graphs by expanding them as sums over pseudo orbits, sets of periodic orbits. Only a finite collection of pseudo orbits which are irreducible and where the total number of bonds is less than or equal to the number of bonds of the graph appear, analogous to a cut off at half the Heisenberg time. The calculation simplifies previous approaches to pseudo orbit expansions on graphs. We formulate coefficients of the characteristic polynomial and derive a secular equation in terms of the irreducible pseudo orbits. From the secular equation, whose roots provide the graph spectrum, the zeta function is derived using the argument principle. The spectral zeta function enables quantities, such as the spectral determinant and vacuum energy, to be obtained directly as finite expansions over the set of short irreducible pseudo orbits.

The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure

In this paper we give a proof of the Gauss–Bonnet theorem of Connes and Tretkoff for noncommutative two tori \mathbb{T}_{\theta}^2 equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number \tau in the upper half plane, representing the conformal class of a metric on \mathbb{T}_{\theta}^2 , and a Weyl factor given by a positive invertible element k \in C^{\infty}(\mathbb{T}_{\theta}^2) , the value at the origin, \zeta (0) , of the spectral zeta function of the Laplacian \triangle\mkern-.5mu ' attached to (\mathbb{T}_{\theta}^2, \tau, k) is independent of \tau and k .

Quantum mechanics on Laakso spaces

We first review the spectrum of the Laplacian operator on a general Laakso space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, we compute the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculate the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the resulting zeta function. Lastly, we derive a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for some special cases

Heat Kernel Coefficients for Laplace Operators on the Spherical Suspension

In this paper we compute the coefficients of the heat kernel asymptotic expansion for Laplace operators acting on scalar functions defined on the so called spherical suspension (or Riemann cap) subjected to Dirichlet boundary conditions. By utilizing a contour integral representation of the spectral zeta function for the Laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure. Thanks to the well known relation between the zeta function and the heat kernel obtainable via Mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions. The particular case of a $d$-dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given explicitly.

Spectral determinants and zeta functions of Schrödinger operators on metric graphs

A derivation of the spectral determinant of the Schrödinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader. To formulate the spectral determinant, we first derive the spectral zeta function of the Schrödinger operator using an appropriate secular equation. The result obtained for the spectral determinant is along the lines of the recent conjecture (Texier 2010 J. Phys. A: Math. Theor. 43 425203).

SPECTRAL ZETA FUNCTIONS OF A 1D SCHRÖDINGER PROBLEM

We study the spectral zeta functions associated to the radial Schrödinger problem with potential V(x) = x2M + αxM-1 + (λ2 - 1/4)/x2. After directly computing some of the zeta functions, we use the quantum Wronskian equation to give sum rules between them, allowing for instances where the explicit form of the zeta functions can be simplified. An immediate application of this work is to derive functional relations and identities involving hypergeometric series, allowing for known identities to be found as instances of more general results. Our work is then extended to a class of related -symmetric eigenvalue problems. Using the fused quantum Wronskian, we give a simple method for indirectly calculating the associated spectral zeta functions. This method is then applied to calculate the nonlocal integrals of motion Gn which appear in an associated integrable quantum field theory.

On Riesz Means of Eigenvalues

In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann–Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5–3.7) and conjecture about additional bounds.

Vacuum energy in conical space with additional boundary conditions

The total vacuum energy of some quantized fields in conical space with additional boundary conditions is calculated. These conditions are imposed on a cylindrical surface which is coaxial with the symmetry axis of conical space. The explicit form of the matching conditions depends on the field under consideration. In the case of the electromagnetic field, the perfectly conducting boundary conditions or isorefractive matching conditions are imposed on the cylindrical surface. For a massless scalar field, the semi-transparent conditions (δ-potential) on the cylindrical shell are investigated. As a result, the total Casimir energy of the electromagnetic field and the scalar field, per unit length along the symmetry axis, proves to be finite, unlike the case of an infinitely thin cosmic string. In these studies, the spectral zeta functions are widely used. It is shown briefly how to apply this technique for obtaining the asymptotics of the relevant thermodynamical functions with a high-temperature limit.

Conical Casimir pistons with hybrid boundary conditions

In this paper, we compute the Casimir energy and force for massless scalar fields endowed with hybrid boundary conditions, in the setting of the bounded generalized cone. By using spectral zeta function regularization methods, we obtain explicit expressions for the Casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston. Our general formulas are, subsequently, specialized to the case in which the piston is modeled by a d-dimensional sphere. In this particular situation, explicit results are given for d = 2, 3, 4, 5.

Spectral zeta function of the sub-Laplacian on two step nilmanifolds

We study the heat kernel trace and the spectral zeta function of an intrinsic sub-Laplace operator ΔL∖Gsub on a two step compact nilmanifold L∖G. Here G is an arbitrary nilpotent Lie group of step 2 and we assume the existence of a lattice L⊂G. We essentially use the well-known heat kernel expressions of the sub-Laplacian on G due to Beals, Gaveau and Greiner. In contrast to the spectral zeta function of the Laplacian on L∖G which can have infinitely many simple poles it turns out that in case of the sub-Laplacian only one simple pole occurs. Its residue divided by the volume of L∖G is independent of L and can be expressed by the Lie group structure of G. By standard arguments this result is equivalent to a specific asymptotic behaviour of the heat kernel trace of ΔL∖Gsub as time tends to zero. As an example we explicitly calculate the spectrum of the sub-Laplacian ΔL∖Gsub in case of the six-dimensional free nilpotent Lie group G and a standard lattice L⊂G by using a decomposition of ΔL∖Gsub into a family of elliptic operators.