Explicit spectral density matrices and spectral types for concrete singular even-order ordinary differential operators: completing the concrete side of Kodaira's 1950 eigenfunction-expansion theory

Kodaira (1950) proved that every self-adjoint realization of a formally self-adjoint ordinary differential expression of arbitrary even order 2ν with real, sufficiently smooth coefficients on an interval with singular endpoints admits a full eigenfunction expansion, whose measure is a ν×ν spectral density matrix ρ^{jk}(λ) obtained by a Stieltjes-type inversion of the characteristic (Weyl) matrix M^{jk}(l). The theorem is abstract: it guarantees existence of ρ for all such operators but computes it for essentially none with genuinely non-constant coefficients. Scope for further work remains in several directions: the coefficient-regularity direction has been absorbed by the modern theory of canonical/Hamiltonian systems with locally integrable coefficients; the complex/non-self-adjoint and odd-order directions survive only as active programs without a universal target (a spectral resolution is impossible in general there); and explicit spectral data have been computed for constant-coefficient powers of the derivative (Bush-Liaw-Martin 2022), for the single fourth-order Bessel-type equation (Everitt-Markett school, 1994-2024), and for eventually constant, periodic, and approximately constant classes. What remains open is the explicit determination of ρ^{jk}(λ) together with the absolutely-continuous/point/singular-continuous spectral decomposition for concrete even-order operators (2ν ≥ 4) with genuinely non-constant coefficients, including the sixth- and higher-order members of the countably infinite Bessel-type family and higher-order analogues of the classical Airy/Laguerre/Jacobi operators.

Background

For second-order Sturm-Liouville operators $-y'' + q(x)y$ on a half-line, Weyl's limit-point/limit-circle theory and its successors (Titchmarsh, Kodaira, Levinson) construct a scalar spectral measure $\rho(\lambda)$ and an expansion theorem: suitable functions decompose as integrals against $d\rho$ over generalized eigenfunctions, with Parseval's identity. This is the abstract foundation of the explicit catalogue of second-order special-function transforms (Fourier, Fourier-Bessel/Hankel, Airy, Laguerre, Jacobi) on which much of mathematical physics rests. Kodaira (1950) extended the abstract theory to arbitrary even order. A 'formally self-adjoint' expression of order $n = 2\nu$ is $L = \sum_{r=0}^{n} p_r(x)(d/dx)^{n-r}$ with real coefficients satisfying the Lagrange symmetry conditions that make $\int \bar f \, Lg \, dx$ symmetric after integration by parts. An endpoint is 'singular' if the coefficients fail to be integrable there or the interval is infinite, so that initial-value data at that endpoint is unavailable; self-adjoint realizations are then selected by abstract boundary conditions, classified up to deficiency indices $(m,m)$, where the 'deficiency index' $m$ counts the dimension of $L^2$-solutions of $Ly = \pm iy$ at a singular endpoint (limit-circle case). Kodaira classifies problems into $(\nu+1)^2$ types via the null planes and excess indices $\tau_a, \tau_b$ of the two endpoints, obtains deficiency $(\tau,\tau)$, constructs the Green's function (norm bounded by $|\operatorname{Im} l|^{-1}$), and builds the 'characteristic matrix' $M^{jk}(l)$ — the higher-order analogue of Weyl's $m$-function — from which the 'spectral density matrix' $\rho^{jk}(\lambda)$ is recovered by the Stieltjes-type double limit $\rho^{jk}(\lambda) = \lim_{\delta \to +0} \lim_{\varepsilon \to +0} \frac{1}{\pi}\int_{\lambda}^{\lambda+\delta} \operatorname{Im} M^{jk}(\mu + i\varepsilon)\,d\mu$. The expansion theorem then diagonalizes every self-adjoint realization, its kernel given by integrals of products of the generalized eigenfunctions against the matrix measure $d\rho^{jk}$. The 'spectral type' of a realization is the decomposition of its spectrum into absolutely continuous, point, and singular continuous parts with multiplicities — the qualitative structure that governs scattering, decay of the associated evolution equation, and inversion. In second order this structure is explicit for the classical special-function operators; in order $2\nu \ge 4$ with non-constant coefficients no fundamental system of solutions is available in closed form, the spectral data are matrix-valued ($\nu \times \nu$, so multiplicity $\ge 2$), and the boundary values of $M^{jk}(l)$ on the real axis become hard asymptotic-analysis objects (Levinson-form asymptotic integration, i.e., constructing fundamental solutions asymptotically in the large-spectral-parameter regime, and dichotomy estimates, which separate exponentially growing from decaying solution modes). Consequently, 75 years after Kodaira's theorem, the explicit spectral density matrix is known for essentially no even-order operator of order $\ge 4$ with genuinely non-constant coefficients.

Problem Statement

Let $L = \sum_{r=0}^{2\nu} p_r(x)(d/dx)^{2\nu-r}$ be a formally self-adjoint ordinary differential expression of even order $2\nu \ge 4$ with real-valued coefficients that are genuinely non-constant in the operator-level sense, i.e. whose coefficients are not all eventually constant, not all periodic, not all piecewise constant (step), and do not all tend to finite limits at an infinite singular endpoint (in particular, the operator is not in the almost-constant class of Ref. 6), and such that $L$ is not one of the operators whose spectral density matrix and spectral type are already established in the cited literature, namely: (i) the constant-coefficient operators $(-1)^{\nu} (d/dx)^{2\nu}$ and their finite-rank perturbations (the Bush–Liaw–Martin class, Ref. 22); (ii) the fourth-order Bessel-type operator $(xy'')'' - ((9x^{-1} + 8M^{-1}x)y')' = \Lambda x y$ on a half-line (Refs 18–21); or (iii) any operator obtained, by an invertible change of the independent variable, a similarity (gauge) transformation by a multiplication operator, or an explicit power or polynomial, from an operator in (i) or (ii) or from any lower-order operator with known spectral data; the solver is not required to prove any general non-reducibility property, and $L$ is admitted unless it is one of the listed operators or is obtained from one of them by one of the listed transformations, satisfying Kodaira's classical smoothness hypotheses (each coefficient $p_m$ has continuous derivatives up to order $2\nu - m$) or their modern locally integrable replacements, on an interval with at least one singular endpoint, and let $T$ be a specified self-adjoint realization (concrete boundary conditions). By Kodaira's theorem, $T$ admits a spectral resolution with a $\nu \times \nu$ spectral density matrix $\rho^{jk}(\lambda)$ obtained from the characteristic matrix $M^{jk}(l)$ by the Stieltjes-type limit $\rho^{jk}(\lambda) = \lim_{\delta \to +0} \lim_{\varepsilon \to +0} \frac{1}{\pi} \int_{\lambda}^{\lambda+\delta} \operatorname{Im} M^{jk}(\mu + i\varepsilon)\, d\mu.$ The problem is to determine, for one concrete such operator, jointly: (a) the spectral density matrix $\rho^{jk}(\lambda)$ in explicit form — a closed form in named special functions, or a representation from which the absolutely continuous, point, and singular continuous decomposition of the spectrum and its multiplicities are directly read off (a numerical or algorithmic evaluation with a stated error bound may be supplied as a supplement, but does not by itself constitute an explicit form) — under stated conventions; and (b) the spectral type of $T$: the absolutely continuous, point, and singular continuous parts of its spectrum, with multiplicities, together with the location of any discrete eigenvalues. Since the spectral type is in principle readable off an explicit $\rho$, (a) and (b) are two facets of one objective: making Kodaira's abstract spectral data concrete. Priority instances: the members of the countably infinite family of even-order Bessel-type equations beyond the developed fourth-order case (sixth and higher; the developed fourth-order member is $(xy'')'' - ((9x^{-1} + 8M^{-1}x)y')' = \Lambda x y$ on a half-line, in the sense of the Everitt school); higher-order analogues of the classical second-order special-function operators (Airy-type, Laguerre-type, Jacobi-type); fourth- and higher-order operators with polynomial or rational coefficients, including confining or unbounded leading behaviour.

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