Direct Spectral Evaluation (DSE)

• Direct Spectral Evaluation (DSE) is a framework that computes spectral data via integral representations, capturing key features like discrete poles and continuum thresholds.
• It employs tailored renormalization and regularization schemes—such as dimensional spectral renormalization and the BPHZ spectral scheme—to manage divergences while preserving underlying symmetries.
• The approach uses both numerical discretization and analytical iteration to rapidly converge on spectral functions, with applications demonstrated in 2+1D φ⁴-theory and indefinite canonical systems.

Direct Spectral Evaluation (DSE) encompasses a class of methods for the direct computation of spectral data—including real-time correlation functions, monodromy matrices, and spectral measures—via integral representations involving spectral variables. DSE frameworks cast equations from quantum field theory, functional integrals, or canonical systems directly into spectral space, enabling the extraction of non-perturbative physical information such as multi-particle continua, discrete poles, and threshold behaviors. The approach is characterized by respect for underlying symmetries and renormalization procedures tailored to the spectral variable domain, and finds applications in both functional approaches to strongly correlated systems and spectral theory for indefinite canonical systems (Horak et al., 2020, Langer et al., 1 Apr 2026).

1. Spectral Formalism and Representations

Central to Direct Spectral Evaluation is the use of spectral (Källén–Lehmann-type) representations, which express n-point functions or operator kernels through integrals over spectral densities. For a scalar field two-point function $G(p^2)$, the representation reads

$$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$

where $\rho(\lambda)$ is the spectral function, encoding delta-peaks for stable particles and a continuum for multi-particle states. Similar spectral representations apply to higher $n$-point vertices: e.g., the 4-point $s$-channel vertex involves a real-valued spectral weight $\rho_4(\mu)$ integrated against a corresponding propagator kernel.

In canonical systems with indefinite Hamiltonians, solutions to first-order ODEs of the form $J y'(x) = z H(x) y(x)$ admit representations through fundamental solutions and boundary value data, with the spectral variable $z$ acting as a spectral parameter (Langer et al., 1 Apr 2026).

2. Transition from Functional/Formal Equations to Spectral DSEs

The DSE approach transforms functional equations—such as Dyson–Schwinger equations in quantum field theory or canonical system boundary conditions—into coupled integral equations over spectral variables. For the $\phi^4$-theory in $2+1$ dimensions, the master equation for the effective action $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$0 produces a hierarchy of DSEs for $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$1-point functions. Upon inserting spectral representations for every propagator and vertex, loop diagrams become multidimensional spectral integrals over analytic kernels, typically computable in closed form after performing momentum integrals via dimensional regularization (Horak et al., 2020).

For indefinite canonical systems, DSE manifests in the assignment of generalized boundary (regularized) values at singularities and the construction of interface conditions and monodromy matrices, directly connecting the analytic and spectral theory of the underlying operators (Langer et al., 1 Apr 2026).

3. Spectral Renormalization and Regularization Schemes

DSE necessitates customized renormalization procedures to deal with divergences emerging in spectral integrals. Two main schemes are employed:

• Dimensional spectral renormalization: The divergence structure is controlled using dimensional regularization, splitting the ultraviolet (UV) tail of the spectral weight into analytic and numerically tractable parts. Counterterms are introduced analogous to standard field-theoretic renormalization, ensuring finite integrals and full preservation of gauge and spacetime symmetries.
• Spectral BPHZ scheme: By Taylor-expanding external momenta under the spectral integrals up to the degree of divergence and subtracting the leading terms (analogous to BPHZ in momentum space), all spectral integrals become convergent. This scheme avoids explicit analytic evaluation of divergent integrals at the price of introducing Taylor subtractions.

In both approaches, all symmetries (e.g., gauge, chiral, spacetime) remain intact, and no symmetry-breaking regulators are required (Horak et al., 2020).

4. Numerical and Analytical Solution Strategies

The DSE formalism yields coupled, nonlinear systems of integral equations for the spectral functions. Solution proceeds via discretization and iterative refinement:

• A grid of spectral masses $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$2 is selected, and initial ansätze (e.g., delta peaks for poles) are posited for the spectral functions.
• Spectral Dyson–Schwinger equations are evaluated numerically over these grids, and iterative updates to the spectral functions are performed according to the imaginary part of analytically continued inverse two-point functions.
• Physical constraints—including on-shell conditions $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$3 and normalization rules such as $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$4—are imposed at each step.
• Convergence is typically rapid (on the order of 10 iterations for standard grids), and the resulting spectral functions display clear signatures of particle poles, multi-particle thresholds, and continuum behavior (Horak et al., 2020).

For canonical systems, the solution involves solving ODEs for fundamental solutions on each side of a singularity and assembling the global solution through the interface (jump) matrix and monodromy factorization, leading directly to spectral data extraction (Langer et al., 1 Apr 2026).

5. Applications: $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$5-Theory and Indefinite Canonical Systems

Direct Spectral Evaluation has been explicitly implemented in the following contexts:

• $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$6-theory in 2+1 dimensions: The spectral function $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$7 of the scalar propagator exhibits a delta pole at the physical mass $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$8, a rising two-particle continuum beginning at $$G(p^2) = \int_0^\infty \frac{d\lambda}{\pi} \frac{\lambda \rho(\lambda)}{p_0^2 + \lambda^2}$$9, and suppressed higher thresholds (3-, 4-particle onsets). Increasing the coupling $\rho(\lambda)$0 accentuates deviations from the free form, and the method effectively captures both single-particle and multi-particle components. The $\rho(\lambda)$1-channel vertex spectral function $\rho(\lambda)$2 reveals similar threshold behaviors, including UV asymptotics and the impact of higher-particle states. Enhanced skeleton expansions incorporating full vertices further improve the physical consistency of the spectral functions near thresholds (Horak et al., 2020).
• Indefinite canonical systems with inner singularities: DSE is used to systematically handle systems with non-integrable singularities in the Hamiltonian. Generalized boundary values are constructed on each side of the singularity, and a 2×2 interface jump matrix is derived, encoding the effect of localized singularity parameters. The fundamental solution (and monodromy matrix) across the interval is factorized into a "continuous" part (bulk Hamiltonian-dependent) and a "discrete" part (depending only on the singularity data). Spectral information, including the Weyl–Titchmarsh $\rho(\lambda)$3-function and spectral measure, is directly extracted from the analytic properties of the monodromy matrix (Langer et al., 1 Apr 2026).

6. Extraction of Spectral Data and Physical Interpretation

DSE ensures that spectral data—such as spectral measures, $\rho(\lambda)$4-functions, and physical observables—are computed directly from the analytic structure of the solution. This includes:

• Recovery of spectral functions as discontinuities of analytically continued correlation functions.
• Extraction of normalization residues for stable poles, with multi-particle continua characterized by support and threshold structure.
• Identification of eigenvalues of self-adjoint realizations via zeros of monodromy matrix entries.
• Determination of the full spectral measure using Herglotz representations based on the constructed transfer and monodromy matrices.

A plausible implication is that DSE frameworks furnish a unifying structure for the computation of non-perturbative information in both quantum field theories and indefinite operator theory, preserving symmetry and analytic properties throughout (Horak et al., 2020, Langer et al., 1 Apr 2026).