Eigenfunction Expansions in Spectral Analysis

• Eigenfunction expansions are a mathematical method that represents functions as sums or integrals of eigenfunctions, ensuring completeness and orthonormality.
• They are applied to differential operators, discrete systems, and quantum mechanics, providing efficient solutions for complex boundary value problems.
• The framework underpins convergence results and spectral analyses, offering insights into error bounds, spectral measures, and operator theory.

Eigenfunction expansions are foundational tools in spectral theory, functional analysis, differential equations, mathematical physics, and numerical analysis. Their central idea is to represent functions or solutions of linear operators as (possibly infinite) sums or integrals of eigenfunctions of those operators. This framework encompasses classical expansions such as Fourier, Sturm–Liouville, and generalized spectral decompositions in abstract settings. Recent developments give comprehensive results for complex operator classes, geometric settings, and applications to abstract, differential, discrete, or operator-valued systems.

1. Fundamental Concepts of Eigenfunction Expansions

An eigenfunction expansion represents a function or solution in terms of a system of eigenfunctions associated with a linear operator, typically arising from a boundary value or spectral problem. Formally, if $A$ is a (possibly unbounded) operator in a Hilbert space $H$ with pure point spectrum and an orthonormal eigenbasis $\{e_j\}$, every $f\in H$ has the expansion

$f = \sum_{j}(f,e_j)_H\,e_j,$

with convergence in the norm of $H$. More generally, when the spectrum is continuous or mixed, expansions involve Plancherel-type direct integrals of generalized eigenfunctions (Lenz et al., 2013).

For differential or difference operators on appropriate domains, the eigenfunctions are solutions of associated homogeneous problems with boundary or initial conditions. The procedure leads to completeness, orthonormality, and Parseval-type identities under appropriate spectral assumptions.

2. Eigenfunction Expansions in Abstract and Concrete Operator Settings

In the abstract Hilbert-space context, let $L$ be a normal or self-adjoint operator with pure point (discrete) or mixed spectrum. The spectral theorem gives a decomposition into direct integrals over spectral measures: $$H \cong \int_{\sigma(L)}^\oplus H(\lambda)\,d\mu(\lambda),$$ where $H(\lambda)$ is a fiber space consisting of generalized eigenfunctions. If the spectrum is pure point, the sum is countable; for continuous spectrum, one obtains integral expansions. The Plancherel identity and completeness hold (Lenz et al., 2013), and unconditional convergence in various Banach norms is addressed through additional regularity or decay conditions (Mikhailets et al., 2023).

Concrete settings include:

• Differential operators (e.g., Sturm–Liouville, elliptic) on domains or manifolds, yielding classical series expansions (Dasgupta et al., 2014, Vučković et al., 2015, Papanicolaou et al., 2023).
• Discrete symplectic systems and finite-difference equations, with expansions in terms of orthogonal eigenvectors on finite or semi-infinite intervals (Zemánek, 2024).
• Boundary-integral operator eigenexpansions in separable geometries, using explicit bases such as spherical harmonics (Bardhan et al., 2012).
• Applications to quantum Hamiltonians with singular or parameter-dependent coefficients, requiring singular Weyl–Titchmarsh theory and operator-valued spectral measures (Smirnov, 2015).

3. Structure, Spectral Measures, and Weyl–Titchmarsh Theory

Eigenfunction expansions often connect to the analytic structure of the associated resolvent or $M$-functions (Weyl–Titchmarsh functions). For example, for discrete symplectic systems with spectral parameter dependence, the expansion theorem expresses arbitrary solutions (or inhomogeneous problems) in terms of an orthonormal basis of eigenfunctions derived from the associated boundary value problem, with Parseval's identity holding in the natural semi-norm (Zemánek, 2024). The Weyl–Titchmarsh $M$-function is an analytic, matrix-valued Nevanlinna function with integral representation over the spectral function: $$M_{N+1}(\lambda) = M^{[0]} + \lambda M^{[1]} + \int_{-\infty}^\infty \left(\frac{1}{t-\lambda}-\frac{t}{1+t^2}\right) dT_{a,B}(t),$$ where $T_{a,B}(t)$ accumulates spectral data (eigenvalues, multiplicities, and eigenfunction vectors). The imaginary part gives the spectral measure, and the expansion is closely linked to the poles and residues of $M(\lambda)$ (Zemánek, 2024).

4. Convergence, Unconditionality, and Function Spaces

A fundamental issue is the convergence type of eigenfunction expansions outside the Hilbert norm. For classical Fourier or elliptic operator eigenfunctions, convergence in $L^2$ is guaranteed. However, convergence in stronger topologies (Sobolev, Hӧrmander, $C^\ell$, or $L^p$ spaces) is more delicate; it is generally not absolute in infinite-dimensional settings. The main results—e.g., (Mikhailets et al., 2023)—give necessary and sufficient conditions (in terms of spectral weights and function space parameters) for unconditional convergence in a Banach space $N$:

• Sufficient and necessary spectral decay rate conditions on eigenvalues (via the function $\alpha(t)$), such that $$\int_1^\infty t^{2\ell+n-1}\alpha^{-2}(t)\,dt<\infty$$ for convergence in $C^\ell$.
• Quantitative error bounds and convergence rates depending on the regularity of $f$ (as measured in refined scales such as Hӧrmander or Sobolev spaces).

Komatsu ultradifferentiable or Gelfand–Shilov spaces admit complete characterizations in terms of eigenfunction coefficient decay, with explicit exponentials for analytic, Gevrey, or broader function classes (Dasgupta et al., 2014, Vučković et al., 2015).

5. Discrete Symplectic Systems: Matrix Difference Equations and Expansion Theorems

Symplectic systems of the form $$Z_k(\lambda) = (S_k + \lambda V_k) Z_{k+1}(\lambda)$$, with boundary data enforced by special matrices, generalize classical discrete Sturm–Liouville theory (Zemánek, 2024). Under nondegeneracy and semi-inner-product structure induced by $V_k \geq 0$, the main expansion theorem asserts: $$z(\lambda) = \sum_{j=1}^r\sum_{i=1}^{m_j} c_{j,i}\, Z^{[i]}(\lambda_j),$$ with explicit Parseval identity and orthonormality in the weighted semi-inner-product. The spectral function $T_{a,B}(t)$ encodes the full spectral information, and the associated Weyl–Titchmarsh $M$-function gives both analytic and measure-theoretic structure, satisfying a Stieltjes integral formula. The theory extends to half-line cases under the appropriate limiting process.

Typical examples include block-diagonal or banded $S_k, V_k$ matrices, with eigenvectors corresponding to physical modes (such as oscillators or quantum systems with internal degrees of freedom).

6. Applications and Illustrative Examples

Eigenfunction expansions are pivotal in:

• Solving PDEs with complex or singular boundary conditions (e.g., sixth-order thin-film operators (Papanicolaou et al., 2023), Schrödinger operators with strong singularities (Smirnov, 2015)).
• Expanding fundamental solutions for classic equations in coordinates adapted to problem geometry (e.g., parabolic or elliptic cylinder coordinates for Laplace's equation (Cohl et al., 2012)).
• Nonlocal or parameter-dependent spectral problems (Nevalinna-type operator relations (Khrabustovskyi, 2012), quantum scattering problems (Sakhnovich, 2018)).
• Discrete and combinatorial settings, such as Laplacians on graphs or spectral theory in metric measure spaces (Lenz et al., 2013).

The systematic framework, as in (Zemánek, 2024), unifies the expansion theory for general linear spectral-parameter dependence, details the spectral/measure connection, and provides templates for further generalizations and concrete analysis.

7. Extensions, Open Problems, and Further Directions

The eigenfunction expansion paradigm is extendable to:

• Operators in Kreĭn spaces or with indefinite or degenerate forms (applicable in Hamiltonian systems and indefinite metric theory (Komech et al., 2013)).
• Systems with nonselfadjoint, matrix, or operator-valued coefficients; nonuniform weights; and boundary conditions dependent on the spectral parameter (via boundary-triplet and Nevanlinna function theory (Mogilevskii, 2020, Mogilevskii, 2013)).
• Tensor-based and sequence-space representations of ultradifferentiable functions and ultradistributions, with precise tensor calculus for operator mappings (Dasgupta et al., 2017).

Contemporary research explores rates of unconditional convergence, asymptotic behavior of spectral measures, interplay with boundary data, and non-classical function space settings on manifolds with complex geometry. The significance of spectral completeness, basisness, and optimal expansion rates for both theoretical analysis and algorithmic implementations is a persistent theme across applications.

References:

• (Zemánek, 2024) — Theory for discrete symplectic systems with spectral parameter dependence
• (Mikhailets et al., 2023) — Abstract unconditional convergence and rates in function spaces
• (Dasgupta et al., 2014, Vučković et al., 2015, Dasgupta et al., 2017) — Spectral characterizations for ultradifferentiable and analytic function spaces
• (Komech et al., 2013) — Hamiltonian systems and Kreĭn-space expansions
• (Mogilevskii, 2020, Mogilevskii, 2013, Khrabustovskyi, 2012) — Nonclassical boundary conditions, operator relations, and expansions
• (Bardhan et al., 2012, Cohl et al., 2012, Papanicolaou et al., 2023) — Explicit expansions in special geometries and high-order boundary value problems
• (Lenz et al., 2013) — Generalized eigenfunction expansions for metric measure spaces and graphs