Darboux–Crum Transformations in Spectral Theory

• Darboux–Crum transformations are algebraic procedures that generate new solutions and exactly solvable operators from spectral problems while preserving spectral properties.
• They enable the construction of exceptional orthogonal polynomials, such as the X_m Laguerre family, through isospectral techniques and careful factorization.
• The commutativity of iterated transformations ensures shape invariance and systematic hierarchies, extending classical Sturm–Liouville frameworks.

The Darboux–Crum transformations constitute a deeply interlinked family of algebraic procedures for generating new solutions and exactly solvable systems from spectral problems, most notably in the context of Sturm–Liouville operators and orthogonal polynomials. In the setting of polynomial Sturm–Liouville problems, these transformations not only yield new families of exceptional orthogonal polynomials—such as the $X_m$ Laguerre polynomials—but also demonstrate isospectrality (spectral preservation) and fundamental structural properties like shape invariance. The permutability or commutativity of iterated Darboux transformations, as codified by the Darboux–Crum theorem, is central to understanding how these transformations produce hierarchies of exactly solvable operators and their associated polynomial solutions, systematically extending the classical frameworks described by Bochner's theorem.

1. Darboux Transformations Adapted to Polynomial Sturm–Liouville Operators

A polynomial Sturm–Liouville (PSL) operator is of the form

$T(y) = p(x)y'' + q(x)y' + r(x)y,$

and is said to be polynomially exactly solvable (PES) if it possesses an infinite ladder of polynomial eigenfunctions. Classical families (e.g., Hermite, Laguerre, Jacobi) are characterized by this property. For the Darboux transformation to preserve the polynomial nature, an "algebraic Darboux transformation" is constructed using a quasi–rational factorization eigenfunction $\phi(x)$ (satisfying $T(\phi) = \lambda_0 \phi$) and a rational gauge $b(x)$, so that

$T - \lambda_0 = B A,$

where $A$ and $B$ are first order differential operators: $$A(y) = b(x)\left(y' - w(x) y\right), \quad B(y) = \hat{b}(x)\left(y' - \hat{w}(x) y\right).$$ The choice of $w(x), \hat{w}(x)$ and $b(x), \hat{b}(x)$ is dictated by the requirement that $A$ (acting on polynomial eigenfunctions) does not introduce poles, ensuring the images are again polynomials.

The partner operator is then defined as

$\widehat{T}(y) = A B y + \lambda_0 y,$

with intertwining relations

$A T = \widehat{T} A, \qquad B \widehat{T} = T B.$

Thus, if $y_n(x)$ is an eigenpolynomial of $T$ with eigenvalue $\lambda_n$, then $A(y_n)$ is an eigenfunction of $\widehat{T}$ with the same eigenvalue. With a suitable gauge, $A(y_n)$ remains a polynomial.

2. Isospectrality, Shape Invariance, and Their Analytical Consequences

When the factorizing eigenfunction $\phi$ is not itself a polynomial, the Darboux transformation becomes isospectral: the spectrum of the new operator $\widehat{T}$ coincides with that of $T$, except possibly omitting the eigenvalue $\lambda_0$ associated with $\phi$. The intertwining ensures that spectral properties are inherited—crucially, there is a canonical way to "remove" or "miss" an index in the polynomial ladder, generating so-called exceptional polynomial families.

Shape invariance is a property fundamental in exactly solvable quantum systems and underlies the existence of ladder operators. For a parametric family $T_k$ (with parameter $k$), shape invariance is the statement that there exists a factorization (with appropriate first–order operators $A_k,B_k$ and constant $\lambda_k$)

$T_k = B_k A_k + \lambda_k,$

and the partner operator can be written as

$$\widehat{T}_k = A_k B_k + \lambda_k = T_{h(k)} + \text{const},$$

with $h(k)$ a discrete shift in parameter space (e.g., $k\mapsto k+1$ for Laguerre). For the Darboux–generated exceptional polynomials, this manifests as explicit raising/lowering operators: $$\hat{A}_{k,m} L^{\text{I}}_{k,m,n}(x) = -L^{\text{I}}_{k+1,m,n-1}(x), \quad \hat{B}_{k,m} L^{\text{I}}_{k+1,m,n}(x) = (n+1-m) L^{\text{I}}_{k,m,n+1}(x).$$ These relations are structurally identical to those satisfied by their classical counterparts, except that the exceptional families "miss" a finite number of lowest degrees.

3. Generation of $X_m$ Laguerre Polynomials via Isospectral Darboux Transformations

To explicitly produce the $X_m$ Laguerre polynomials, the classical Laguerre operator

$L_k(y) = x y'' + (k+1 - x) y'$

is factorized using an appropriate nonpolynomial eigenfunction at shifted parameter and a rational gauge: $$\phi(x) = e^x L_{k,m}(-x), \quad \tilde{L}_{k,m}(x) = L_{k,m}(-x),$$ and operators

$$A_{k,m}(y) = \tilde{L}_{k,m}(x) y' - \tilde{L}_{k+1,m}(x) y, \qquad B_{k,m}(y) = \frac{x y' + (k+1) y}{\tilde{L}_{k,m}(x)}.$$

The transformed operator has as eigenfunctions objects of the form $A_{k,m}(L_{k, n}(x))$, which are rational multiples of classical Laguerre polynomials, beginning at degree $m$—thereby constructing the exceptional $X_m$ Laguerre family. Analogous constructions apply for the type II exceptional family, utilizing alternative factorizations.

4. Permutability of Darboux–Crum Transformations and Its Structural Implications

A central structural insight is the permutability or commutativity of iterated Darboux transformations: performing $n$ successive Darboux steps, each with possibly distinct factorization functions $\phi_1,\ldots,\phi_n$, leads to a higher–order intertwiner expressible in terms of a Wronskian determinant,

$$A(y) = b(x)\frac{W[\phi_1,\ldots,\phi_n,y]}{W[\phi_1,\ldots,\phi_n]},$$

where $b(x)$ is a gauge factor. Crucially, the resulting higher–order operator is independent (up to sign) of the order in which the elementary Darboux steps are performed. The permutability ensures that the final partner operator depends only on the set of factorization seeds, not their sequence. Algebraically, for intertwining operators $B$ and $T_0,T_n$,

$T_0 B = B T_n,$

where $B = B_1 B_2 \ldots B_n$.

This property is essential for shape invariance, as it guarantees that the successive application of Darboux transformations preserves the functional form (up to parameter shift) of the operator, ensuring the existence of higher–order raising and lowering operators in the exceptional family.

5. Structure and Properties of Exceptional Orthogonal Polynomials

The exceptional ($X_m$) orthogonal polynomials arising from isospectral Darboux–Crum transformations exhibit several critical features:

• Gaps in the spectrum: The polynomial eigenfunctions do not begin at degree zero but start at degree $m$; lower degrees are absent.
• Retained orthogonality: With appropriate weight functions (e.g., $\tilde{L}_{k,m}(x) e^{-x} x^k$ for the $X_m$ Laguerre case), the exceptional polynomials remain orthogonal.
• Bispectrality and recurrence: Exceptional families satisfy second–order differential equations (of Sturm–Liouville type), but their three-term recurrence relations are replaced by higher-order recurrences, reflecting the absence of lower degrees.
• Shape invariance: The algebraic structure (commutative diagram of parameter-shifted operators and their accompanying eigenfunctions) persists due to the permutability of the Darboux–Crum transformations.

The exceptional orthogonal polynomial families constructed via this framework extend the classical Bochner classification and exhibit highly nontrivial algebraic and analytic properties, including completeness with respect to their modified weights and the existence of explicit raising/lowering operators.

6. Summary Table: Algebraic Structure of Darboux–Crum Transforms for $X_m$ Laguerre Polynomials

Step	Formula/Operator	Key Principle/Outcome
Darboux factorization	$T - \lambda_0 = B A$	Algebraic factorization using quasi–rational function
Partner operator	$\widehat{T}(y) = A B y + \lambda_0 y$	Isospectral partner, intertwining relations
Iterated transform	$$A(y) = b(x) W[\phi_1, \ldots, \phi_n, y]/W[\phi_1, \ldots, \phi_n]$$	Permutability, independence from order
Raising/lowering	See equations (16), (17) in the main text	Shape invariance, ladder operators
Weight function	Modified classical weight (e.g., involving $\tilde{L}_{k,m}$)	Orthogonality of exceptional polynomials

7. Broader Implications and Connections

The adaptation of Darboux–Crum transformations to polynomial Sturm–Liouville operators, and in particular to the construction of exceptional ($X_m$) orthogonal polynomials, demonstrates the far-reaching scope of these methods. The isospectrality and shape invariance derived from the permutability structure underlie the algebraic and spectral completeness of exceptional families and offer a systematic generalization of classical, exactly solvable models. These results create pathways to new developments in spectral theory, representation theory, and exactly solvable models in mathematical physics, expanding the landscape of orthogonal polynomials and integrable systems well beyond the traditional families dictated by Bochner's theorem.