Spectral Zeta Function

• Spectral zeta function is a complex analytic tool that encodes the eigenvalues of self-adjoint operators using series and heat kernel transforms.
• It achieves meromorphic continuation and reveals pole structures that link spectral properties to geometric invariants and quantum field applications.
• The method underpins zeta-regularized determinants, bridging analytic techniques in number theory, spectral geometry, and mathematical physics.

A spectral zeta function is a function that encodes the spectral data (eigenvalues) of a linear operator—typically a self-adjoint, positive, unbounded operator on a Hilbert space—into a complex analytic object suitable for regularization, analysis, and extraction of geometric and physical invariants. The construction is foundational across mathematical physics, spectral geometry, number theory, and quantum field theory, and forms the analytic backbone of determinant regularization, spectral actions, and identification of various analytic and arithmetic structures.

1. Definition and Fundamental Properties

For a self-adjoint operator $A$ with discrete, positive spectrum $\{\lambda_k\}_{k=1}^\infty$, the standard spectral zeta function is defined as

$$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$

for $\Re(s)$ large enough to ensure absolute convergence. This function is generally extended to a meromorphic function on $\mathbb{C}$ using analytic continuation techniques. In geometrically or physically motivated settings (e.g., Laplace‐type operators on compact manifolds, quantum mechanical Hamiltonians, combinatorial Laplacians, Dirac operators on metric graphs, or operators on fractal or $p$-adic spaces), the spectral zeta function is often representable as a Mellin transform of the heat trace: $$\zeta_A(s) = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1}\ \operatorname{Tr}\left(e^{-tA}\right) dt$$ The same structure extends to more general eigenproblems, with shifted and Hurwitz-type zeta regularizations, and to non-commutative or infinite-dimensional situations. The analytic structure of $\zeta_A(s)$ (location of poles, residues) is determined by the short-time heat trace asymptotics.

2. Meromorphic Continuation and Analytic Structure

The principal analytic feature of spectral zeta functions is meromorphic continuation. For broad classes of operators (e.g., elliptic on compact manifolds), the zeta function extends to the entire complex plane with at most simple poles at a discrete set of points. The pole structure is controlled by the Weyl expansion and Seeley–DeWitt/Minakshisundaram–Pleijel coefficients from the heat kernel short-time expansion. In the case of regular Sturm–Liouville or Laplacian-type operators, techniques such as the Liouville-Green (WKB) expansion, contour-integral representations, and subtraction of asymptotic terms yield explicit analytic continuations, pole locations, and values of residues in terms of spectral, geometric, or combinatorial data (Fucci et al., 2013, Fucci et al., 2021, Fucci et al., 2011). In exotic settings, such as quasi-regular Sturm–Liouville operators, branch point singularities may appear instead of mere poles (e.g., at $s=0$ for certain Bessel-type operators, or at each non-positive integer for Legendre-type operators) (Fucci et al., 2024).

3. Spectral Zeta Functions in Diverse Geometric and Quantum Systems

Sturm–Liouville and Schrödinger Operators

Regular and quasi-regular Sturm–Liouville operators, including Schrödinger-type systems and their self-adjoint extensions, admit fully explicit spectral zeta analyses. The key methodology involves expressing the zeta function as a contour integral around the eigenvalues, with the integrand related to the logarithmic derivative of a characteristic function constructed from boundary data or Green's functions. Special values at integer $s$ are computed via recursion from the small-$\{\lambda_k\}_{k=1}^\infty$0 expansion of the characteristic determinant (Fucci et al., 2013, Fucci et al., 2021, Fucci et al., 2024). For radial Schrödinger operators, closed-form expressions for low moments of the spectral zeta (e.g., $\{\lambda_k\}_{k=1}^\infty$1) can be given in terms of hypergeometric and Meijer $\{\lambda_k\}_{k=1}^\infty$2-functions, and connections to nonlocal integrals of motion in integrable models are derived through the quantum Wronskian framework (Watkins, 2011).

Quantum Graphs

On finite metric graphs (quantum graphs), the spectral zeta function of operators such as the Dirac operator is formulated using determinant expressions via the argument principle, leading to explicit contour-integral representations. Analytic continuation is achieved by subtracting the appropriate small- and large-$\{\lambda_k\}_{k=1}^\infty$3 asymptotics of the secular determinant. The regularized zeta-determinants yield explicit combinatorial expressions involving the metric data (e.g., bond lengths) and matching (scattering) matrices (Harrison et al., 2016).

Graph Laplacians and Discrete Models

For discrete graphs, both finite (e.g., cyclic or torus graphs) and infinite (e.g., $\{\lambda_k\}_{k=1}^\infty$4 lattices, regular trees), the combinatorial Laplacian’s spectral zeta function acquires a structure parallel to the classical Riemann zeta function. Explicit representations in terms of beta and hypergeometric integrals are obtained, and “completed” zeta functions satisfy functional equations analogous to those in analytic number theory. The analytic structure critically reveals deep connections to the Riemann hypothesis, as detailed for families of graph zeta functions whose asymptotics encode information equivalent to the Riemann or Epstein–Riemann conjectures (Friedli et al., 2014, Meiners et al., 2022, Karlsson, 2019).

Fractals and $\{\lambda_k\}_{k=1}^\infty$5-adic Systems

Spectral zeta functions admit generalization to fractal Laplacians (e.g., on the Sierpinski gasket) and $\{\lambda_k\}_{k=1}^\infty$6-adic Laplacians. In these cases, the spectral zeta function can be factored into contributions from geometric zeta functions, dynamical zeta functions linked to renormalization or iteration maps, and hyperfunction factors (e.g., Dirac delta hyperfunctions). The analytic properties often reflect the underlying self-similarity or $\{\lambda_k\}_{k=1}^\infty$7-adic scaling, with the location and nature of singularities (e.g., infinitely many poles or branch cuts) deviating strongly from the Archimedean case (Lal et al., 2012, Chacón-Cortés et al., 2015).

4. Functional Determinants and Regularization

A central application is the regularization of determinants of differential (or pseudodifferential) operators. The zeta-regularized determinant is defined as

$\{\lambda_k\}_{k=1}^\infty$8

providing a well-defined prescription for the determinant of an unbounded operator with infinitely many eigenvalues. In quantum field theory and spectral geometry, zeta regularization underlies the computation of effective actions, vacuum energies, partition functions, and local geometric invariants. For certain operators on graphs or quantum graphs, this determinant is directly expressible in terms of combinatorial data and exhibits relations to Mahler measures and arithmetic invariants (Harrison et al., 2016, Kurkov et al., 2014).

Beyond the standard derivative prescription, finite-difference (or $\{\lambda_k\}_{k=1}^\infty$9-deformed) spectral zeta regularization generalizes the determinant notion, probing the operator spectrum at two distinct values and connecting to Tsallis entropies and information-geometric structures, thus introducing a parametrized hierarchy of non-extensive spectral aggregators (Okamura, 13 Apr 2026).

5. Applications in Mathematical Physics and Quantum Field Theory

Spectral zeta functions are universally employed for analytic continuation and regularization of path integrals, one-loop effective actions, and partition functions in quantum field theories on curved backgrounds. The analytic extraction of local counterterms and identification of anomalies (e.g., conformal anomalies) follow directly from pole structures and heat-kernel coefficients. At the multi-loop level, generalized Barnes-type zeta functions control overlapping and nested divergences, providing a fully covariant renormalization scheme (Bilal et al., 2013, Kurkov et al., 2014).

In exactly solvable models, such as the quantum Rabi model, the spectral zeta function encodes the interplay between the operator’s spectral data and universal limiting objects. In the deep strong-coupling regime, the spectral zeta function for the quantum Rabi model converges to the classical Hurwitz/Riemann zeta function, thus establishing a spectral universality (Hiroshima et al., 2024).

Special values of the spectral zeta function appear in explicit path-integral computations via Feynman–Kac-type representations, facilitate probabilistic constructions of ground-state measures for spin-boson and field-theoretic models, and are recoverable as moments of quantum observables (Hiroshima et al., 2024, Kurkov et al., 2014).

6. Connections to Number Theory, Special Functions, and Functional Equations

Spectral zeta functions often realize, generalize, or approximate structures familiar from analytic number theory. Functional equations for graph zeta functions, spectral zeta functions on lattices, and connection-Laplacian zetas on simplicial complexes mirror the analytic properties of the Riemann zeta and $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$0-functions, sometimes including completed forms and self-reciprocity. In combinatorial and geometric settings, special values of the spectral zeta function align with Catalan numbers, Dedekind sums, or appear in the evaluation of Verlinde formula dimensions (Knill, 2018, Karlsson, 2019).

For graphs and their Laplacians, analytic continuation and completion lead to functions satisfying exact or approximate self-reciprocal functional equations, and encode, via their asymptotics, reformulations of the Riemann or generalized Riemann (Epstein) hypotheses (Friedli et al., 2014, Meiners et al., 2022). In one-dimensional simplicial complexes, the spectral zeta of the squared connection Laplacian exhibits an explicit involutive functional equation, with spectral symmetry under inversion (Knill, 2018).

Hypergeometric and special-function connections are pervasive: Appell $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$1, Lauricella $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$2, and beta integral representations occur in the analytic continuation of zeta functions on graphs and trees, and the analytic structure may be encoded in explicit closed-form hypergeometric representations (Friedli et al., 2014, Watkins, 2011).

7. Methodologies for Analytic Continuation, Special Values, and Residues

The standard toolkit for the analysis of spectral zeta functions includes:

• Contour-integral / argument principle methods: Representation of the zeta function as a logarithmic derivative of a secular or characteristic function around the eigenvalues, suitable for both ordinary and partial differential operators (Fucci et al., 2013, Fucci et al., 2024, Harrison et al., 2016).
• WKB and Liouville-Green expansions: Extraction of large eigenvalue / large spectral parameter asymptotics to perform subtractions and enable meromorphic continuation (Fucci et al., 2013, Fucci et al., 2011).
• Heat-kernel and Mellin transforms: Use of the short-time heat expansion to identify pole locations and residues, with direct link to geometric invariants (volume, curvature, etc.) (Stern, 2017, Fucci et al., 2013).
• Spectral determinant small-parameter expansion: Systematic expansion of the characteristic determinant near $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$3, yielding explicit recursions for the values $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$4 at positive integers (Fucci et al., 2021, Fucci et al., 2024).
• Probabilistic and path-integral techniques: Especially in quantum models, spectral traces are related to expectations over stochastic processes, with the zeta encapsulating statistical moments and correlations (Hiroshima et al., 2024).
• Finite-difference regularization and information geometry: The $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$5-deformation paradigm interpolates zeta regularization families and encodes novel nonextensive aggregators and induced metrics (Okamura, 13 Apr 2026).

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References cited:

• (Fucci et al., 2013) Spectral Functions for Regular Sturm-Liouville Problems
• (Fucci et al., 2021) Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators
• (Fucci et al., 2024) The spectral $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$6-function for quasi-regular Sturm--Liouville operators
• (Fucci et al., 2011) The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type $$\zeta_A(s) = \operatorname{Tr}(A^{-s}) = \sum_{k=1}^{\infty} \lambda_k^{-s}$$7
• (Harrison et al., 2016) Zeta Functions of the Dirac Operator on Quantum Graphs
• (Karlsson, 2019) Spectral zeta functions
• (Friedli et al., 2014) Spectral zeta functions of graphs and the Riemann zeta function in the critical strip
• (Watkins, 2011) Spectral zeta functions of a 1D Schrödinger problem
• (Chacón-Cortés et al., 2015) Heat Traces and Spectral Zeta Functions for p-adic Laplacians
• (Lal et al., 2012) Hyperfunctions and Spectral Zeta Functions of Laplacians on Self-Similar Fractals
• (Stern, 2017) Finite-rank approximations of spectral zeta residues
• (Okamura, 13 Apr 2026) Finite-difference zeta function regularisation and spectral weighting in effective actions
• (Hatsuda, 2015) Spectral zeta function and non-perturbative effects in ABJM Fermi-gas
• (Kurkov et al., 2014) Spectral action with zeta function regularization
• (Bilal et al., 2013) Multi-Loop Zeta Function Regularization and Spectral Cutoff in Curved Spacetime
• (Hiroshima et al., 2024) Spectral zeta function and ground state of quantum Rabi model
• (Meiners et al., 2022) Spectral zeta function on discrete tori and Epstein-Riemann conjecture
• (Knill, 2018) An Elementary Dyadic Riemann Hypothesis