Spectral Zeta Functions: Theory & Applications

• Spectral Zeta Functions are analytic functions defined as Dirichlet series over eigenvalues of differential operators, capturing geometric and physical invariants.
• They utilize techniques like heat kernel asymptotics, Mellin transforms, and Poisson resummation to perform analytic continuation and regularize divergent sums.
• These functions effectively bridge spectral theory and applications such as Casimir effect regularization, Bose–Einstein condensation, and geometric analysis.

Spectral zeta functions (SZFs) are analytic functions constructed from the spectrum of (typically unbounded) differential operators, encoding both geometric and analytical properties of the underlying domains and operators. In their prototypical form, an SZF is defined as a Dirichlet series over the nonzero eigenvalues $\{\lambda_n\}$ of a self-adjoint (partial) differential operator $P$, for example,

$\zeta_P(s) = \sum_n \lambda_n^{-s}$

where $s \in \mathbb{C}$ is a complex parameter. SZFs originated in spectral geometry and mathematical physics for the systematic regularization of divergent sums and the extraction of geometric and physical invariants from spectral data. These functions have become fundamental tools across mathematics and physics, with applications ranging from the analysis of geometric boundary problems and quantum field regularization to the paper of phase transitions and spectral geometry.

1. Analytical Framework and Construction

The foundations of SZFs lie in associating to $P$ (often Laplace type) its spectrum and forming the aforementioned Dirichlet series. For elliptic operators on compact manifolds, the set $\{\lambda_n\}$ is discrete and positive (after suitable removal of zero modes when necessary). The analytic continuation of $\zeta_P(s)$ from its domain of convergence to the entire complex plane is established via the integral representation

$$\zeta_P(s) = \frac{1}{\Gamma(s)}\int_0^\infty t^{s-1} K(t)\,dt \tag{∗}$$

where $K(t) = \mathrm{tr}(e^{-tP})$ is the heat trace. The small-$t$ asymptotic expansion of $K(t)$ as

$K(t) \sim \sum_{k} a_k\, t^{k-d/2}$

(where $d$ is the spatial dimension) determines the pole structure and residues of $\zeta_P(s)$ via the Mellin transform, and thus encodes geometric invariants such as volume and boundary contributions. Techniques such as Poisson resummation and contour integration are employed to access alternative representations, enlarge the region of convergence, and facilitate analytic continuation.

2. Key Examples: Hurwitz, Epstein, and Barnes Zeta Functions

Three central families of SZFs arise naturally in spectral problems:

• Hurwitz Zeta Function:

$\zeta_H(s, q) = \sum_{n=0}^{\infty}(n+q)^{-s}$

(for $\mathrm{Re}(s)>1$, $0<q\leq 1$), typically for spectra linear in $n$ and relevant to one-dimensional oscillators and strings.

• Epstein Zeta Function:

$$Z_d(s) = \sum_{n\in \mathbb{Z}^d \setminus \{0\}} Q(n)^{-s}$$

where $Q$ is a positive-definite quadratic form; naturally appears in periodic domains, tori, and problems with multidimensional periodic boundary conditions.

• Barnes Zeta Function:

$$\zeta_B(s, c|\vec{f}) = \sum_{m_1,\ldots,m_d=0}^{\infty} [c + m_1 f_1 + \ldots + m_d f_d]^{-s}$$

a multidimensional generalization, relevant for higher-dimensional oscillatory spectra such as the harmonic oscillator in Bose–Einstein condensation problems.

All these examples share the property that their series representations arise from the spectrum of suitable (partial) differential operators, with the zeta function constructed as a sum over the eigenvalues dictated by the geometry and boundary conditions of the underlying domain (Kirsten, 2010).

3. Analytic Continuation and Residues

Central to the analytical theory of SZFs is their meromorphic continuation and the extraction of residues, which are explicitly linked to geometric features of the underlying space. The integral formula (∗) connects the meromorphic structure of $\zeta_P(s)$ to the heat kernel's small-time asymptotics. Poles of $\zeta_P(s)$ reflect geometric invariants such as volume, boundary area, and curvature integrals. Tools for analytic continuation include:

• Writing spectral sums as contour integrals via the use of the complex Gamma function and Cauchy's theorem, exploiting representations of $\lambda^{-s}$ as Laplace transforms.
• Poisson resummation to recast sums over eigenvalues as rapidly convergent dual series, particularly useful for multidimensional tori (Epstein zetas).
• Residue calculus to relate the coefficients in the asymptotic expansion of $K(t)$ to the positions and residues of the poles of $\zeta_P(s)$.

By this machinery, one can define regularized determinants via $\log\det_\zeta P = -\zeta_P'(0)$ and regularize otherwise divergent physical expressions, such as the Casimir energy or partition sums in quantum statistical mechanics.

4. Applications in Mathematical Physics

SZFs have achieved prominence as practical tools in the analysis of quantum and statistical physical systems:

• Casimir Effect: The quantum vacuum energy in confined geometries is a formally divergent sum $\frac12\sum_n \omega_n$ over eigenfrequencies $\omega_n$. Replacing this by the spectral zeta prescription gives the regularized energy as $E_\text{Cas} = -\frac12 \zeta_P'(0)$. The dependence of the Casimir force on boundary geometry and position is accessed by studying how the SZF encodes geometric information (Kirsten, 2010).
• Bose–Einstein Condensation (BEC): For a trapped Bose gas, the critical temperature $T_c$ and thermodynamic quantities are determined by the eigenvalue spectrum of the trapping potential's Schrödinger operator. The grand canonical partition function is naturally rewritten in terms of spectral zeta functions; asymptotic expansions at high temperature (small $\beta$) reveal how boundary and geometric effects influence $T_c$ and thermodynamic behavior.

In both these contexts, SZFs bridge microscopic spectral data and macroscopic observables, with physical manifestations directly related to analytic properties, such as pole residues and their associated heat kernel coefficients.

5. Geometric Information and "Hearing the Shape"

SZFs provide a conduit for accessing geometric invariants from spectral data. This is epitomized in questions such as "Can one hear the shape of a drum?", where one probes whether the geometric form of a manifold is encoded in the eigenvalue spectrum of its Laplacian. The small-$t$ expansion of the heat trace—and thus the pole structure of $\zeta_P(s)$—contains contributions corresponding to area, perimeter, and more refined invariants such as curvature integrals, but in general does not determine the geometry uniquely. The spectral invariants extracted from the residues of the SZF serve as a geometric fingerprint, establishing which physical and mathematical quantities are "audible" to the spectrum (Kirsten, 2010).

6. Analytical Techniques and Unified Mechanism

The paper emphasizes that all commonly encountered zeta functions (Hurwitz, Epstein, Barnes) can be constructed via a unified "mechanism": each arises from Dirichlet series over eigenvalue spectra of (possibly parametrized) partial differential operators with given boundary data. Their analytic properties, asymptotic expansions, and the nature of their pole structure all reflect the operator and the domain. Key techniques span contour integration, asymptotic series expansions, Poisson summation, residue calculus, and the use of special function identities linking zeta functions to the underlying geometry (Kirsten, 2010).

7. Motivating Questions and Broader Context

SZFs stand at the intersection of spectral theory, geometry, and analytic number theory. They are uniquely positioned to address and clarify questions such as:

• What geometric properties of a domain are encoded in the spectrum of a differential operator?
• To what extent do quantum and statistical phenomena (vacuum fluctuations, phase transitions) "know" about the geometry and topology of the underlying space?
• How does the interplay between spectrum and geometry manifest in physical observables, e.g., the force in the Casimir effect or the thermodynamics of BEC?

Through the analytic structure of $\zeta_P(s)$—in particular its poles and residues—one can derive and interpret quantitative relationships between geometric measures and physical effects, highlighting the profound utility of the spectral zeta function in mathematical physics, quantum theory, and geometric analysis.

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In summary, spectral zeta functions serve as a powerful analytical tool to connect the spectrum of differential operators to the underlying geometry and physical phenomena. Their construction via eigenvalue sums, analytic continuation, and residue analysis provides a unifying language to access geometric invariants, regularize divergent sums, and express physical observables in a mathematically rigorous framework—demonstrating their central role in both the mathematics and physics of spectral theory (Kirsten, 2010).