Spectral Expansion Problem

• Spectral Expansion Problem is the representation of functions or dynamics as sums or integrals over eigenfunctions of a linear operator, defined by its spectrum.
• It involves methodologies like biorthogonal systems and principal value integration to manage non-self-adjoint and nonlocal operators.
• Applications span quantum mechanics, stochastic processes, and computational mathematics, highlighting challenges in analytic completeness and numerical convergence.

The spectral expansion problem concerns the possibility and methodology of representing functions, solutions, or dynamics associated with a linear operator as superpositions of eigenfunctions, indexed either discretely or continuously, with expansion coefficients determined by the structure of the underlying operator and the associated spectrum. In mathematical physics, operator theory, and applied mathematics, spectral expansions underpin the analysis of differential, difference, and integral operators—both self-adjoint and non-self-adjoint, in settings ranging from finite-dimensional spaces and Hilbert spaces to spaces of distributions and stochastic dynamical evolutions. The literature documents a diverse set of frameworks for spectral expansion, tailored to operator class (self-adjoint, non-self-adjoint, nonlocal), spectrum structure (discrete, continuous, mixed), and functional analytic context (Schwartz distributions, stochastic processes, discrete symplectic systems, etc.), as well as the challenges of practical computation and analytical completeness.

1. Foundational Concepts and Classical Results

At its core, a spectral expansion represents a function or solution as a sum or integral over normalized eigenfunctions of a linear operator, each weighted by a spectral coefficient encoding the function's projection onto corresponding eigenspaces. In separable Hilbert spaces, the spectral theorem for self-adjoint operators ensures such expansions in terms of orthonormal bases, with Parseval's equality guaranteeing $L^2$ convergence and energy conservation. Generalizations—such as the Schwartz expansion for tempered distributions (Carfí, 2011), the eigenfunction expansion for discrete symplectic systems with general spectral parameter dependence (Zemánek, 21 Dec 2024), or spectral decompositions in the context of stochastic master equations (Vinci et al., 14 Aug 2025)—extend these principles beyond classical settings to operators acting in broader spaces or with more intricate spectral structures.

Classical spectral expansions for self-adjoint operators take the form

$$A(u) = \sum_{j} \lambda_j \langle u, e_j \rangle e_j$$

for discrete spectra, and as integrals for continuous spectra or when the space is enlarged to include generalized eigenvectors (e.g., via the Gelfand–Shilov formalism).

2. Extensions to Non-Self-Adjoint and Nonlocal Operators

Many operators of interest in modern applications are non-self-adjoint and may possess complex-valued, non-real spectra, or lack an orthonormal eigenfunction basis. For differential or difference operators with periodic, possibly complex coefficients, the spectral expansion problem involves additional technical complications: the spectrum may not be purely real, spectral singularities (ESS, SQ) may preclude naive termwise integration, and Parseval's identity typically fails (Veliev, 23 Sep 2025).

For such operators, spectral expansions often rely on biorthogonal systems $(\Psi_n, \Psi_n^*)$ and Riesz bases, and expansions may be expressed as

$$f(x) = \frac{1}{2\pi} \sum_n \int_{-\pi}^{\pi} a_n(t) \Psi_{n, t}(x) \, dt$$

with $a_n(t)$ determined by inner products with the dual system. The existence of essential spectral singularities generally requires grouping (and principal value integration) of problematic terms to ensure convergence and analytic meaningfulness (Veliev, 23 Sep 2025).

For operators associated with stochastic processes (Fokker–Planck, master equation), the spectral expansion is not over spatial eigenfunctions but over eigenmodes of the generator, reflecting both relaxation rates and observable dynamics. In open or time-dependent systems, the expansion coefficients evolve according to non-autonomous ODEs or, more generally, through operator-valued coupling terms (Vinci et al., 14 Aug 2025).

3. Continuous Superposition: From Summation to Integration

A key generalization is the replacement of summation over discrete spectra by integration over continuous spectral parameters. In the theory of Schwartz linear operators acting on spaces of tempered distributions, the spectral expansion becomes

$A(u) = \int_{\mathbb{R}^m} a(p)[u|v](p) v_p \, dp$

where $[u|v](p)$ is the coordinate distribution with respect to a S-linearly independent Schwartz eigenfamily $v$ indexed by $\mathbb{R}^m$ (Carfí, 2011). This mirrors the Dirac formalism common in quantum mechanics: $$A\,|u\rangle = \int a(p)\,|p\rangle\langle p|u\rangle\, dp$$ and rigorously realizes formal manipulations prevalent in physics within distribution theory.

Expansions of this type arise as well in the finite-temperature spectral expansion of two-point functions in integrable quantum field theories, where thermal traces are replaced by sums over eigenstates, converted into multidimensional contour integrals via residue calculus (Szécsényi et al., 2012).

4. Biorthogonal Systems, Spectral Operators, and Completeness

For non-self-adjoint operators, the lack of an orthonormal basis necessitates the use of biorthogonal systems. Completeness of such systems, and thus validity of the spectral expansion, is contingent on the structure of the spectrum and absence of spectral singularities. The inner product structure is often semi-definite or defined with respect to operator-dependent weights or spectral functions, especially in discrete symplectic systems (Zemánek, 21 Dec 2024). The Weyl–Titchmarsh M-function, which admits an integral representation in terms of the spectral function $T$, plays a central role in encapsulating the spectral data and enables linking the spectral measure to the expansion coefficients.

An archetypal formula for the M-function is

$$M_{n+1}(\lambda) = M[0] + \lambda M[1] + \int_{-\infty}^\infty \left[ \frac{1}{t-\lambda} - \frac{t}{1 + t^2} \right] dT_{a, B}(t)$$

with the jumps of $T_{a, B}$ corresponding to eigenvalues, and the residues dictating the contribution of each eigenfunction.

5. Spectral Expansion in Stochastic and Dynamical Systems

In statistical physics and stochastic processes, the spectral expansion formalism is leveraged to solve moment-closure problems and analyze relaxation dynamics. By expanding the probability density $p(x, t)$ in the eigenbasis $\{\phi_n(x)\}$ of the Fokker–Planck operator, and introducing a dual adjoint system, one can project the solution onto the space of observables (moments), yielding a matrix ODE for moment evolution (Vinci et al., 14 Aug 2025): $\frac{d}{dt} m(t) = G_\lambda(t) [ m(t) - m_*(t)]$ with $G_\lambda = U \Lambda U^{-1}$, $U_{k n} = \int f_k(x) \phi_n(x) \, dx$, and $\Lambda$ the diagonal matrix of eigenvalues. This formalism automatically captures system time scales and enables analytical closure even for non-Gaussian systems, including Bessel processes or nonlinear mean-field models.

Connections to the Koopman operator and data-driven modal decomposition underscore the broad scope of spectral expansions for system reduction and analysis.

6. Applications, Challenges, and Computation

Spectral expansion methodologies underlie practical and theoretical advances in quantum mechanics (expansion over scattering states, spectral theory of Schrödinger operators), statistical physics (moment closure for open systems with time-dependent parameters), and applied mathematics (eigenfunction expansions for boundary value problems with transmission or parameter-dependent conditions (Mukhtarov et al., 2013), or discrete settings such as symplectic systems (Zemánek, 21 Dec 2024)). The development of algorithms for certifying spectral expansion properties, such as for expander graphs using Chebyshev polynomials and geodesic cycle counts (Huang, 2019), attests to the utility of spectral expansion in computational mathematics and network theory.

Key challenges include:

• Non-self-adjointness, leading to absence of Parseval-type identities and potential spectral singularities (Veliev, 23 Sep 2025)
• Nonintegrability or divergence of expansion coefficients, requiring grouping and principal value integration
• Dealing with open systems, where the spectral basis itself may vary with time or parameters, introducing additional coupling terms (Vinci et al., 14 Aug 2025)
• Ensuring completeness and biorthogonality in weighted or semi-definite inner product spaces
• Numerical computation in high dimensions, where constructing the spectral basis or approximating integrals over the continuous spectrum becomes nontrivial

In stochastic and quantum systems, contemporary methods also leverage spectral expansion techniques for analytic continuation (e.g., stochastic pole expansions or Nevanlinna analytic continuation for reconstructing QCD spectral functions (Huang et al., 2023)), and in high-dimensional combinatorics (such as expansion phenomena on simplicial complexes and local-to-global spectral gap descent (Oppenheim, 2014, Oppenheim, 2017, Oppenheim, 2018)).

7. Summary Table of Operator Classes and Spectral Expansion Features

Operator Class / Problem Setting	Type of Expansion	Key Features / Complications
Self-adjoint (Hilbert)	Discrete/continuous sum, orthonormal	Parseval, spectral theorem, strong theory
Non-self-adjoint (periodic coeff., complex)	Fourier–Floquet, biorthogonal	ESS, grouping, principal value integrals
Schwartz linear operators (tempered dist.)	Continuous integral (indexed by ℝ^m)	S-linear independence, generalized functions
Discrete symplectic (general parameter dep.)	Finite sum, semi-inner product	Weyl–Titchmarsh M-function, extension to ∞
Stochastic generators (Fokker–Planck/master eq.)	Eigenmode expansion, moments	Non-Hermitian, coupling in open systems
Expander graphs, combinatorial Laplacians	Chebyshev/Cycle-based, moments	Explicit algorithmic certification

This table encapsulates the diversity of spectral expansion problems and the corresponding solution frameworks developed in the literature.

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In conclusion, the spectral expansion problem is a cornerstone across numerous mathematical and physical domains, with contemporary research providing rigorous generalizations, analytic tools for challenging operator classes, and algorithmic methods for complex, high-dimensional, or non-self-adjoint settings. The continued refinement of these frameworks enables both deeper theoretical insights and the robust application of spectral methods in applied analysis, quantum mechanics, stochastic dynamics, and discrete mathematics.