Exact Sum Rules (ESRs) in Quantum & Spectral Theory

• Exact Sum Rules (ESRs) are algebraic and analytic identities that relate spectral observables via polynomial and integral constraints emerging from symmetry, analyticity, and integrability.
• They are derived using spectral zeta functions and determinants, which enable precise analytic predictions in quantum mechanics, nuclear physics, and field theory.
• ESRs serve as critical tools for model discrimination, parameter reduction, and validation of numerical methods in diverse systems from quantum oscillators to many-body interactions.

Exact Sum Rules (ESRs) are algebraic or analytic identities—frequently polynomial in content—relating spectral or physical observables in a constrained manner, arising from underlying symmetry, analyticity, or integrable structures. ESRs are found throughout quantum mechanics, spectral theory, nuclear physics, and field theory, and encode deep hidden relations between eigenvalue spectra, zeta functions, and response functions. They serve as fundamental constraints, enable analytic computations, and are pivotal in model discrimination and reduction of parameter space across theoretical and physical contexts.

1. Formulation and Classification of ESRs

ESRs are typically formulated as identities among spectral zeta functions (SZFs)

$$\zeta_n(s) = \sum_{\alpha \in \mathbb{N}_0} E_{n,\alpha}^{-s}$$

where $E_{n,\alpha}$ are the eigenvalues of a quantum mechanical operator—often a Schrödinger or Helmholtz operator—indexed by a "fusion" or sector label $n$. ESRs can take the form of:

• Algebraic sum rules: polynomial relations among $\zeta_n(s)$ (e.g., for fixed $n$, $\zeta_n(s)$ is given by a sum over products of lower-order $\zeta_n(t)$).
• Functional constraints: explicit identities mapping SZFs of different fusion sectors, called zeta generating formulas (ZGFs).
• Moment-constrained integral sum rules: e.g., in spectral theory or response function analysis, relating moments of a spectral density to equal-time expectation values or commutators.

In some cases, ESRs express factorization conditions on spectral determinants, or encode cancellation of divergences (e.g. zero-mode subtractions in trace formulas for inhomogeneous systems (Amore, 2013, Amore, 2019)).

ESRs arise in a variety of contexts, including:

• Quantum oscillators and anharmonic potentials (via exact WKB or ODE/IM methods) (Voros, 2022, Kamata, 8 Aug 2025)
• Quantum billiards, strings, and drums (traces over Green’s functions, diagrammatic expansions, and analytic regularizations) (Amore, 2013, Amore, 2013, Amore, 2017, Amore, 2019, Amore, 2019)
• Nuclear transition strengths (sum rules in the shell model, commutator methods) (Lu et al., 2017, Johnson et al., 2020)
• Response theory (Kubo formulas, generalized Kramers-Kronig relations, nonlinear response) (Bradlyn et al., 24 Apr 2024)
• Quantum field theory and QCD condensate analysis (integral sum rules relating spectral moments and operator expectation values) (Gubler et al., 2016, Gubler et al., 2017, Gubler et al., 2018).

2. Mathematical Foundations: Zeta Functions and Determinants

The algebraic structure of ESRs is anchored in the analytic behavior of spectral zeta functions and their generating spectral determinants: $$\mathcal{D}(E) = \prod_\alpha (E_\alpha - E) = \exp\left( -\zeta'(0) - \sum_{s\geq1} \frac{\zeta(s)}{s} E^s \right)$$ Expanding the determinant in powers of $E$ produces coefficients expressed as polynomials in the $\zeta(s)$, leading to ESRs by equating the coefficients in various sectors or under analytic continuation (Kamata, 8 Aug 2025).

In integrable settings, such as the ODE/IM correspondence, the fusion relations for the so-called T-system: $$C^{(1)}(E) C^{(n)}(\omega^{n+1}E) = C^{(n-1)}(\omega^{n+2}E) + C^{(n+1)}(\omega^nE)$$ map directly, via spectral determinant expansions, to recurrence ESRs on the $\zeta_n(s)$ and to ZGFs connecting different $n$ (Kamata, 8 Aug 2025). These structures are tightly controlled by the symmetry of the problem (e.g., cyclic $\mathbb{Z}_{2M+2}$ or associated Chebyshev polynomial structure).

Semiclassical methods such as exact WKB yield bilinear functional equations for determinants—analyzed by Borel summation and resurgent analysis—again producing ESRs as polynomial constraints on spectral zeta invariants (Voros, 2022).

3. ESRs in Quantum Mechanics and Spectral Theory

In quantum mechanics, ESRs manifest as explicit sum rules for oscillator strengths, eigenvalue moments, and transition probabilities. For instance:

• For one-dimensional quantum oscillators $H_N = -d^2/dq^2 + |q|^N$, the spectral zeta functions $Z_N(s)$ and their combinations (e.g., via parity) satisfy exact WKB-derived polynomial relations:
  • For $N=2$, $Z_2(2m) = \frac{(2\pi)^{2m}|B_{2m}|}{2(2m)!}$
  • For general $N$, higher-order ESRs take the form:

  $$-\cot(\nu\pi) \sin(2n\nu\pi) Z_N^{\#}(n) + \cos(2n\nu\pi) Z_N(n) = \mathcal{P}_{N,n}\{Z_N^{\#}(m)\}_{1\leq m<n}$$

  where $\nu = 1/(N+2)$ and $\mathcal{P}_{N,n}$ is a polynomial in lower $Z_N^{\#}(m)$ (Voros, 2022).

• For perturbed quantum systems (e.g., boxes with delta impurities), ESRs can be calculated explicitly via Green’s function methods and encode the critical couplings for appearance of new bound states through the loci of poles in $Z(s)$ (Amore, 2020).

Diagrammatic expansion techniques, as in inhomogeneous strings, relate ESRs to integrals over products of Green’s functions, with a factorial count of independent diagrams and robust acceleration via Shanks transformation (Amore, 2013). For rational-order sum rules, perturbative expansions and trace-based formalisms yield explicit ESRs, subject to renormalization in the presence of zero modes (Amore, 2019, Amore, 2019, Amore, 2013).

4. ESRs in Many-Body and Nuclear Physics

Transition sum rules in nuclear shell-model theory are prototypical ESRs. Moments of transition strength functions are given by:

• NEWSR (non-energy-weighted): $$S_0 = \langle i | \mathcal{F}^\dagger \mathcal{F} | i \rangle$$

• EWSR (energy-weighted): $$S_1 = \frac{1}{2}\langle i | [\mathcal{F}^\dagger,[H,\mathcal{F}]] | i \rangle$$ which can be decomposed into one- and two-body operator contributions. Their ratio yields the transition centroid and is a key diagnostic for collective phenomena and shell effects (Lu et al., 2017, Johnson et al., 2020).

ESRs thereby provide parameter-free, model-independent benchmarks for the total strength and average energy of transition distributions and serve as a crucial test of many-body computational approximations against configuration-interaction (FCI) benchmarks (Johnson et al., 2020).

Hierarchy sum rules for oscillator strengths—spanning discrete and continuum spectra—unify the Thomas-Reiche-Kuhn rule, generalized Kramers relations, and the virial theorem under a common ESR umbrella, and are validated by extensive numerical calculation (Sukumar, 2018).

5. ESRs in Response Theory and Field Theory

Sum rules for response functions extend ESR concepts to linear and nonlinear response in condensed matter and quantum field theory:

• Linear response sum rules (e.g., $f$-sum rule for conductivity) relate frequency moments of spectral densities to ground-state commutators.

• For second-order response functions, ESRs are derived from a generalized Kubo formula recast in terms of a double spectral density:

  $$\chi^{(2)}_{AB}(\omega_1,\omega_2) = \frac{1}{\pi^2} \int d\alpha\,d\beta\ \frac{\chi^{(2)\prime\prime}_{AB}(\alpha,\beta)}{(\alpha-\omega_{12}^+)(\beta-\omega_2^+)}$$

  • Moments of $\chi^{(2)\prime\prime}_{AB}(\alpha,\beta)$, via integration by parts, are shown to correspond exactly to ground-state equal-time commutators or derivatives (Bradlyn et al., 24 Apr 2024).
  • In solids, the generalized $f$-sum rules for nonlinear optical response can be cast directly in terms of matrix elements of the Bloch Hamiltonian, clarifying the link with quantum geometry and topology.
  • For nonlinear rectification and second-harmonic generation, large-frequency decay is fully determined by these ESRs.

In QCD and finite-temperature field theory, ESRs are integral constraints that relate operator-product expansion (OPE) coefficients (condensates), hydrodynamic parameters (transport coefficients), and moments over the finite-temperature subtracted spectral density: $$(2/\pi) \int_0^\infty \frac{d\omega}{\omega} \delta\rho(\omega) = \mathcal{F}(\langle\text{condensates}\rangle, \text{transport coeffs})$$ These ESRs play a foundational role in constraining lattice spectral reconstructions and extracting quantities such as electrical conductivity, diffusion constants, and relaxation times, without reliance on spectral model assumptions (Gubler et al., 2016, Gubler et al., 2017, Gubler et al., 2018).

6. Algebraic Structures, Symmetries, and Selection Rules

Underlying many ESRs are hidden symmetry structures (e.g., cyclic, fusion, or rotational symmetries). The ODE/IM correspondence clarifies that:

• Fusion relations among Stokes multipliers (quantization functions) correspond algebraically to recurrence ESRs among SZFs (Kamata, 8 Aug 2025).
• ESRs and ZGFs (mapping SZF sets between sectors) are governed by selection rules; only certain orders $s$ yield independent constraints, depending on cyclic subgroup structure (e.g., $\mathbb{Z}_{2M+2}$) determined by the complex-rotational Symanzik symmetry and Chebyshev polynomial realization in the analytic continuation of solutions.
• The mapping between constant parts of Stokes multipliers and Chebyshev polynomials ensures exact factorization properties and classification of ESRs via group-theory.

These symmetries not only generate the ESRs but also ensure invertibility of ZGFs in admissible sectors and determine the dimensionality of independent SZFs at a given order.

7. Significance, Applications, and Outlook

ESRs provide powerful, model-independent constraints and analytic handles for:

• Determining spectral properties and bound-state thresholds via pole structure of sum rules.
• Benchmarks for numerical and approximation methods (Rayleigh–Ritz, FCI, Hartree–Fock, etc.).
• Discrimination among competing physical models (e.g., flavor symmetry models in neutrino physics (Barry et al., 2010), or QCD in-medium behavior (Gubler et al., 2018)).
• Calibrating and constraining spectral reconstructions in lattice QCD, nonlinear response experiments, and quantum geometry measurements in complex materials (Gubler et al., 2016, Bradlyn et al., 24 Apr 2024).
• Revealing hidden connections between quantum mechanical systems, integrable field theories, and algebraic geometry (e.g., via ODE/IM correspondence, Chebyshev and other functional relations).

Future research may focus on extending ESR frameworks to more general classes of potentials and operators, developing higher-order and multi-variable ESRs in nonlinear and many-body settings, and exploiting ESRs for discovery of new integrable structures or for more efficient spectroscopic inference in complex systems.