Trace–Path Integral Formulae & Applications

• Trace–path integral formulae are expressions that represent spectral invariants as integrals over path spaces, unifying operator theory with analytic methods.
• They extract spectral data such as eigenvalues and resonances through analytic continuation and the regularization of determinants using Blaschke products.
• These formulae find applications in quantum scattering, inverse spectral problems, and non-commutative integration, impacting modern mathematical physics.

A trace–path integral formula expresses spectral, dynamical, or quantum invariants (such as traces of operators or monodromy actions) as integrals or sums over suitably defined path spaces. In mathematical physics, such formulae bridge the operator-theoretic and path-integral viewpoints, encoding spectral data or probability amplitudes in terms of integrals over classical or generalized trajectories. They have also found analogues in non-commutative integration, arithmetic geometry, and geometric analysis. Trace–path integral formulae reveal deep connections between complex analysis, spectral theory, stochastic processes, operator algebras, and quantization.

1. Trace Formulas and Complex Analysis for Schrödinger Operators

A representative and technically complete development is given for the 3D Schrödinger operator $H = -\Delta + V(x)$ with real, short-range $V$ (Isozaki et al., 2011). Here, the classical trace formula connecting the spectrum of $H$ to integrals along the real axis is obtained via the analytic structure of the associated Fredholm determinant. The core formulae involve the determinant

$D(k) = \det (I + Q_0(k)),$

where $Q_0(k)$ is constructed from $|V|^{1/2}$ and the free Green's function. The function $D(k)$ is not of trace class, so a Blaschke product $B(k)$ is introduced to yield a modified determinant $D_B(k) = D(k) B(k)$ with "good" analytic properties. The resulting asymptotics,

$$-i \log D(k) = -\frac{\alpha_0}{k} - \frac{\alpha_1}{k^3} + O(1/k^4),$$

imply explicit trace formulas of the form

$$\frac{1}{\pi} \int_{-\infty}^\infty \log |D(t)| dt = -\alpha_0.$$

These encode spectral invariants (e.g., sum of eigenvalues) in logarithmic integrals of the determinant, or, via the Birman–Krein identity,

$\det S(k) = \frac{D(k)}{D(-k)},$

relate the scattering phase to the logarithm of $D(k)$. The central mathematical technique is the recasting of operator spectral data as properties of analytic (entire) functions in the upper half-plane, yielding a duality between trace-type and phase-type formulas.

2. Resonances, Eigenvalues, and Spectral Data Encoded by Zeros

Trace–path integral formulae tightly connect operator spectra (eigenvalues, resonances) to the zero set of complex-analytic functions. In the Schrödinger operator context (Isozaki et al., 2011), zeros of $D(k)$ in the upper half-plane correspond to negative eigenvalues (bound states), and zeros in the lower half-plane to scattering resonances. The trace formula

$$\text{Tr}~f(H) - f(H_0) = \sum_{j=1}^N f(\lambda_j) + \text{resonant~contributions}$$

arises from contour integration around zeros of $D(k)$, extracting spectral sums from the analytic structure of the determinant. The scattering matrix and its phase,

$\varphi_{sc}(k) = ka_1 + \operatorname{arg} D(k),$

are thereby fully determined by the paths in the complex plane (encoded analytically), unifying operator-theoretic and path-analytic perspectives.

3. Dirichlet Integrals, Scattering Phase, and Hardy Spaces

The Dirichlet integral of $\log D_B(k)$ over the upper half-plane,

$$\int_{\mathbb{C}_+} |\nabla \log D_B(k)|^2 \, d^2k + S_0 = m_B,$$

is interpreted as a harmonic function conjugate to the scattering phase. The function $\log |D_B(k)|$ acts as the harmonic conjugate (via the Hilbert transform) to $\arg D_B(k)$. This deep connection, rooted in Hardy space analysis for entire functions, links trace-type (real part) and phase-type (imaginary part) invariants of the system. The theory ensures that knowledge of boundary values (along $\mathbb{R}$) suffices to reconstruct the analytic function $D_B(k)$, and thus all spectral data.

4. Operator-Theoretic versus Path/Function-Theoretic Framework

Traditional trace formulas derive from operator-theoretic arguments (spectral shift, Fredholm determinant, commutator expansions). The analytic approach introduced in (Isozaki et al., 2011) shifts the focus to entire and Hardy class functions. Removing the spectral zeros via the Blaschke product, one produces functions in $H(\mathbb{C}_+)$ that admit both trace (real) and phase (imaginary) Hilbert transforms, generalizing well-known one-dimensional results (e.g., for KdV) to higher dimensions. This approach, beyond unifying trace and phase information, permits direct application to resonance theory, scattering, and integrable quantum systems, and provides machinery to efficiently relate path-integral expressions to operator traces.

5. Applications in Scattering, Inverse Problems, and Integrable Systems

Trace–path integral formulae derived from the analytic formulation play major roles:

• Quantum scattering theory: The methods yield alternative derivations and extensions of fundamental results such as Levinson’s theorem (relating phase shifts to bound state counting), and express the scattering phase entirely in terms of analytic/gauge-invariant boundary data.
• Inverse spectral/quantum problems: By encoding all spectral data in analytic functions determined by potential $V(x)$, one may, in principle, reconstruct $V$ from knowledge of $D(k)$ or the scattering phase—crucial for inverse scattering theory.
• Integrable systems: For the one-dimensional KdV, –log|D(k)| is the action variable; similar scalar quantity relationships for higher-dimensional analogs are suggested.
• Resonance asymptotics and nonlinear equations: The method supports estimation and localization of resonances in non-self-adjoint (or non-Hermitian) problems, with implications for long-term dynamics and stability in quantum, semiclassical, and PDE settings.
• Bridging operator and complex function theory: This duality allows tools from analytic function theory to enter the analysis and solution of spectral and scattering problems in mathematical physics.

6. Broader Context and Methodological Impact

The analytic trace–path integral paradigm is not limited to Schrödinger operators. Similar strategies appear in:

• Non-commutative integration: Trace formulae on crossed-product von Neumann algebras relate modular analytic data to traces on “dual” spaces, using boundary values of analytic interpolators (Yamagami, 2017).
• Geometric and arithmetic settings: Trace–path formulas arise in index theory (as traces over loop or path spaces on manifolds), and even in arithmetic quantum field theory, with path integrals over torsion points of Jacobians equated to traces of Frobenius actions (Cheng, 4 Sep 2025).
• Scattering and field theory: Path-integral methods for heat kernels, anomalies, and partition functions express traces (of exponential semigroups) as integrals over spaces of curves, with boundary contributions linked to trace anomalies and index densities (Bastianelli et al., 2017).

The operational core shared across these domains is the systematic replacement of operator-theoretic traces with integrals or sums over “paths”—in various senses—with (possibly regularized) weights carrying analytic, spectral, or dynamical information. Analytic structure, symmetry, and complex function theory underpin the transition between the two viewpoints.

7. Summary of Canonical Formulas

Some canonical trace–path integral expressions central to this approach include:

Formula Type	Expression	Context/Interpretation
Spectral trace formula	$$\frac{1}{\pi}\int_\mathbb{R}\log|D(t)|dt = -\alpha_0$$	Schrödinger spectral invariant
Dirichlet integral	$$\int_{\mathbb{C}_+} |\nabla \log D_B(k)|^2 d^2k + S_0 = m_B$$	Harmonic conjugate/scattering phase
Scattering phase	$\varphi_{sc}(k) = ka_1 + \arg D(k)$	Connected via Birman–Krein to $D(k)$
Resonance counting	$D(k)$ zeros in $\mathbb{C}_+$ (eigenvalues), $\mathbb{C}_-$ (resonances)	Spectrum via analytic zeros
Non-commutative trace	$$T(f^* f) = \int_\mathbb{R} \langle f(t-i/2), f(t-i/2) \rangle dt$$	Boundary analytic data trace

These formulas codify the deep interplay between traces (as operator-theoretic quantities), path (or configuration) spaces, complex analytic methods, and physical observables.

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Trace–path integral formulae thus form a central conceptual and technical framework bridging analytic, spectral-theoretic, and geometric perspectives across quantum theory, operator algebras, and even arithmetic geometry. Modern developments exploit their dual (operator-function) nature to analyze spectral invariants, resonances, scattering phenomena, anomalies, and quantization beyond the reach of traditional operator theory.