Four-Trace Formula Overview

• Four-trace formula is a spectral identity linking eigenvalue sums of fourth-order operators to arithmetic and geometric invariants across automorphic settings.
• It employs resolvent trace methods and contour integration techniques to derive explicit identities and capture boundary corrections and nonlocal effects.
• Its applications range from inverse spectral problems and spectral counting laws to nonvanishing L-value results, offering actionable insights in spectral theory and number theory.

The four-trace formula is a spectral identity connecting sums over automorphic representations or spectral data to arithmetic/geometric invariants for fourth-order (or generally, order four) operators and automorphic settings. Its manifestations range from explicit spectral trace identities for differential operators to deep identities in the theory of automorphic forms, particularly for cubic-level automorphic forms and higher-rank trace formula constructions. The terminology “four-trace formula” most commonly refers to the resolvent-trace expansions for fourth order differential operators—on domains such as intervals or the circle—that encode the first four moments or traces of spectral shifts in relation to boundary data and potentials. In recent research, it also encompasses higher-rank Kuznetsov or Arthur–Selberg trace formulas for groups like GSp(4) and GL(4), which exhibit rich arithmetic and representation-theoretic structure.

1. Fourth-Order Trace Formulas: Fundamental Structure

For scalar fourth-order differential operators, the core trace formula relates spectral data (eigenvalues) under various boundary conditions to the coefficients of the differential expression and its boundary data. For instance, for the periodic setting on the circle with operator

$$H_2 = \frac{d^4}{dx^4} + 2(p(x)\frac{d}{dx})' + q(x)$$

with 2-periodic boundary conditions, the trace formula reads

$$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$

where $\lambda_n^\pm$ are the band edges for the periodic problem and $\mu_n(x)$ are Dirichlet-type eigenvalues for shifted intervals. The sum converges absolutely, facilitates inverse spectral results, and mirrors the Hill operator trace formulas for second-order operators, but with intricately higher regularity and potential terms (Badanin et al., 2013).

In the interval case with Dirichlet-type boundary conditions, the first few traces encode boundary values and mean values of the potential, e.g.,

$$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$

for $$\Delta_n = \lambda_n - \pi^4 n^4 - p_0\pi^2 n^2 - q_0$$ (Badanin et al., 2014). Higher-order traces involve combinations of $p''(x)$, $p^2(x)$, and associated boundary terms.

2. Derivation and Analytic Techniques

The derivation of four-trace formulas universally employs contour integration techniques, particularly the resolvent trace method. For a higher-order operator $L = L_0 + Q$ (with $L_0$ unperturbed and $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$0 a perturbation), the spectral shift is encoded via

$$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$1

where the Green's kernel expansion and fine analysis of the residue structure produce explicit trace identities. For even-order operators $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$2, a striking phenomenon emerges: when the perturbation $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$3 is singular (possessing, e.g., an atom at the symmetric midpoint), the trace formula acquires an additional nonlocal term:

$$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$4

where $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$5 signifies the strength of the atom at $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$6 and $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$7 is a boundary-dependent constant (Galkovskii et al., 2019). For regular potentials, this term vanishes and one recovers classical Gelfand–Levitan-type results.

3. Arithmetic and Automorphic Contexts: Relative and Kuznetsov Trace Formulas

For automorphic forms, the analogs of the four-trace formula are realized in identities connecting sums of Hecke eigenvalues (and associated $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$8-values) to arithmetic data such as Kloosterman sums and Bessel functions. In the cubic-level setting for $$q(x) - p''(x) = \lambda_0 + \sum_{n=1}^\infty [\lambda_n^- + \lambda_n^+ - 2\mu_n(x)],$$9, this takes the form: $\lambda_n^\pm$0 with explicit diagonal terms and Kloosterman–Bessel sums (Pi et al., 2019). The methodology relies on the relative trace formula, with a test function projecting onto forms of predetermined level and weight, and double unipotent integration explicitly extracting products of Fourier (Whittaker) coefficients. The geometric side involves analysis of orbital integrals of two types (identity and long Weyl orbits), which recapture the spectral sums and arithmetic functions.

For $\lambda_n^\pm$1, the effective Kuznetsov formula constitutes a higher-rank four-trace formula: $\lambda_n^\pm$2 where $\lambda_n^\pm$3 denotes Fourier coefficients, and various $\lambda_n^\pm$4-indexed arithmetic sums (Kloosterman type) and oscillatory integrals appear (Comtat et al., 24 Feb 2025). This identity bridges the spectral and arithmetic sides for automorphic forms on $\lambda_n^\pm$5, enabling representation-theoretic counting and equidistribution results.

4. Applications to Spectral Theory and Arithmetic

Four-trace formulas have several profound applications:

• Inverse Spectral Problems: The trace formula for fourth-order operators uniquely determines potentials $\lambda_n^\pm$6 and $\lambda_n^\pm$7 from spectral data, establishing bijectivity of the spectral map and stability estimates (Badanin et al., 2013).
• Arithmetic $\lambda_n^\pm$8-value Nonvanishing: The cubic-level relative trace formula provides exact moment computations of central $\lambda_n^\pm$9-values, leading to nonvanishing results for $\mu_n(x)$0 in families of modular forms (Pi et al., 2019).
• Spectral Counting Laws: For both automorphic and classical operator settings, the four-trace formula enables explicit Weyl laws quantifying the asymptotic number of forms or eigenvalues in prescribed spectral windows (e.g., $\mu_n(x)$1 growth for Maass newforms at cubic level) (Pi et al., 2019).
• Boundary Correction and Nonlocal Effects: In the presence of singular potential perturbations, the extra term in the trace for even-order operators captures nonlocal interaction between the midpoint singularity and the boundary—the absence of such a term in odd-order cases sharply distinguishes the even-order spectral behavior (Galkovskii et al., 2019).
• Orthogonality and Distribution in Automorphic Families: The four-trace (Kuznetsov) formula for $\mu_n(x)$2 yields quasi-orthogonality of Fourier coefficients, enables large-sieve bounds, and provides information on low-lying zeros of $\mu_n(x)$3-functions, leading to explicit symmetry-type classifications in the sense of Katz–Sarnak (Comtat et al., 24 Feb 2025).

5. Higher-Rank and Generalizations: GL(4) and Beyond

In representation-theoretic analysis, Arthur’s coarse trace formula for $\mu_n(x)$4, while not dubbed “four-trace” in the spectral sense, is a fundamental identity decomposing both geometric and spectral sides into regular and ramified (singular) orbits and discrete/continuous spectra. After truncation, the trace identity reads: $\mu_n(x)$5 where $\mu_n(x)$6 is an orbital (possibly regularized) integral and $\mu_n(x)$7 is a trace or integral over Eisenstein series data (Wang et al., 29 Aug 2025). This framework undergirds the stabilization and comparison of trace formulas across groups and levels, encoding deep functorial and endoscopic correspondences.

6. Summary Table: Key Four-Trace Formulas

Context	Formula Structure	Reference
4th-order operator, circle	$\mu_n(x)$8	(Badanin et al., 2013)
4th-order, interval (Dirichlet)	$\mu_n(x)$9	(Badanin et al., 2014)
Even-order, measure perturbation	$$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$0	(Galkovskii et al., 2019)
$$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$1 cubic-level “Fourier trace”	Sum over $$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$2 of $$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$3 times Hecke eigenvalues equals combinations of Kloosterman sums, Bessel functions	(Pi et al., 2019)
$$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$4 Kuznetsov formula	Weighted sum of Fourier coefficients $$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$5 diagonal + triple Kloosterman sum/arithmetic integral terms	(Comtat et al., 24 Feb 2025)
$$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$6 Arthur–Selberg (coarse)	$$\sum_{n=1}^\infty \Delta_n = \frac{1}{4}[q(0) + q(1) - 2q_0]$$7, matching spectral and geometric sides, after truncation	(Wang et al., 29 Aug 2025)

7. Distinguishing Features, Open Directions, and Significance

The four-trace formula distinguishes itself from second-order (Hill/Gelfand–Levitan) analogs by its enriched structure: boundary term dependence, nonlocal contributions for singular perturbations, and intricate spectral correspondences in automorphic settings. These trace identities serve as powerful tools for spectral synthesis, arithmetic distribution results, and functorial transfers. The rigorous normalization of Whittaker newforms and orbital integrals, as well as explicit regularization for singular orbits, make four-trace formulas central in analyzing spectral and arithmetic phenomena in both mathematical physics and modern number theory (Badanin et al., 2013, Badanin et al., 2014, Galkovskii et al., 2019, Pi et al., 2019, Comtat et al., 24 Feb 2025, Wang et al., 29 Aug 2025).