Exceptional Hermite Polynomials (XHPs)

• Exceptional Hermite polynomials are finite-gap deformations of classical Hermite polynomials obtained via state-deleting Darboux transformations, forming a complete orthogonal system.
• They are constructed using finite sequences of Darboux and Crum transformations, yielding modified weights, higher-order recurrence relations, and determinantal representations tied to integrable systems.
• They enable the creation of exactly solvable quantum models with monodromy-free potentials, providing shape-invariant ladder operators and explicit spectral frameworks.

Exceptional Hermite polynomials (XHPs) are finite-gap deformations of the classical Hermite orthogonal polynomials in which a finite set of degrees is omitted, yet the family forms a complete orthogonal system in a weighted $L^2$ space with respect to an explicitly modified weight. XHPs arise as eigenfunctions of rational extensions of the quantum harmonic oscillator potential via repeated Darboux or Darboux–Crum transformations, and they are fundamentally connected to monodromy-free Schrödinger operators, bispectrality, special function theory, and integrable systems such as the KP hierarchy and Calogero–Moser systems.

1. Construction via Darboux and State-Deleting Transformations

Exceptional Hermite polynomials are constructed by applying finite sequences of state-deleting Darboux transformations to the standard harmonic oscillator eigenproblem. Concretely, let $H_n(x)$ denote the classical Hermite polynomial of degree %%%%2%%%%, and let $X = (\lambda_1,\ldots,\lambda_\ell)$ be a partition, which determines the indices of the eigenstates to be deleted. The associated Wronskian is

$$H_X(x) = \mathrm{Wr}[H_{k_1}(x),H_{k_2}(x),\ldots,H_{k_\ell}(x)],$$

where $k_i = \lambda_i + i - 1$. The rational extension of the potential is then

$U(x) = x^2 - 2\frac{d^2}{dx^2}\log H_X(x).$

The exceptional Hermite polynomials, indexed by $n$ not in the set of deleted degrees, are defined by

$$P_{X,n}(x) = \mathrm{Wr}[H_{k_1}(x),\ldots,H_{k_\ell}(x),H_{n+\ell}(x)].$$

The corresponding eigenfunction is $\psi_n(x) = P_{X,n}(x)e^{-x^2/2}/H_X(x)$.

The resulting set $\{P_{X,n}\}_{n \notin K_X}$, where $K_X$ is the finite set of deleted indices, forms an exceptional orthogonal family (XHPs), which exist only for even codimension $2m$—that is, only for partitions of $m$.

2. Orthogonality, Completeness, and Weight Structure

The XHPs are orthogonal with respect to the weight

$W_X(x) = \frac{e^{-x^2}}{(H_X(x))^2},$

with orthogonality holding provided the Wronskian has no real zeros (i.e., $X$ is an “even” partition). Precisely, $P_{X,n}$ and $P_{X,m}$ are orthogonal if $n \neq m$ in the non-excluded indices, so

$$\int_{-\infty}^{\infty} P_{X,n}(x)P_{X,m}(x) W_X(x)\, dx = c_n\, \delta_{n,m}.$$

The completeness of XHPs in $L^2(\mathbb{R}, W_X(x) dx)$ is established via spectral and monodromy analysis. For each partition $X$, the codimension of the XHPs is $2m = |X|$, and the missing degrees correspond precisely to the set of “exceptional” indices dictated by the Darboux seeds.

An alternative proof of completeness utilizes the trivial monodromy property of the underlying rational potential: the corresponding Schrödinger operator

$$L_X = -\frac{d^2}{dx^2} + x^2 - 2\frac{d^2}{dx^2} \log H_X(x)$$

is monodromy-free (every solution is meromorphic near each singularity), and this ensures that the polynomial eigenfunctions together with the weight $W_X(x)$ span a dense subset of $L^2(\mathbb{R})$ (Gomez-Ullate et al., 2013, Gomez-Ullate et al., 2019, Gomez-Ullate et al., 2020).

3. Differential and Recurrence Properties

Each XHP family is the set of polynomial eigenfunctions of a second-order Fuchsian differential operator

$$T_X[y](x) = y''(x) - 2 \left( x + \frac{H_X'(x)}{H_X(x)} \right) y'(x) + \lambda_n y(x),$$

with $\lambda_n = 2(n - m)$ (for a codimension $2m$ partition $X$).

Unlike the classical case, where Hermite polynomials satisfy a three-term recurrence, XHPs satisfy higher-order recurrences. For a partition of length $\ell$, the recurrence has order $2\ell+3$: $$\sum_{k=-r}^{r} a_{n,k} P_{X,n+k}(x) = X(x) P_{X,n}(x), \quad r = \ell,$$ where the $a_{n,k}$ are rational in $n$ (Durán, 2014, Gomez-Ullate et al., 2015). The minimal order of such recurrences is conjectured to be $2\ell+3$ for XHPs indexed by partitions of length $\ell$.

There exists a bispectral anti-isomorphism between the ring of differential operators in $x$ and the ring of difference operators in $n$, realized via a stabilizer structure that generalizes the spectral duality of the classical Hermite system (Gomez-Ullate et al., 2015, Kasman et al., 2020).

4. Ladder Operators, Shape Invariance, and Polynomial Heisenberg Algebras

XHPs admit the construction of higher-order ladder operators closing into polynomial Heisenberg algebras (PHAs). For example, in codimension $m$, one can construct creation and annihilation operators $b^\dagger$, $b$ of (at least) order $m+1$ such that

$$[H, b^\dagger] = 2b^\dagger, \qquad b^\dagger b = Q(H), \qquad Q(H) = \prod_{j=1}^{m+1}(H - E_j),$$

for certain factorization energies $E_j$ associated with the deleted states (Marquette et al., 2012, Marquette et al., 2012, Grundland et al., 2022).

The appearance of multiple types of ladder operators is a signature of the SUSYQM-based construction, where different chains (Crumm, Krein–Adler) yield alternative operator sets, each associated with a specific algebraic and spectral structure. The Hamiltonians constructed via such rational extensions often inherit shape-invariance of higher order, which enables the determination of the entire spectrum algebraically.

5. Determinantal and Calogero–Moser Representations

XHPs can be expressed via determinantal (Wronskian or Vandermonde-like) formulas:

• For the $X_2$ Hermite family associated to partition $\{1,1\}$, explicit integral and determinantal formulas are given (Simanek, 2022), for example,

$$H_{2,n}^{(\{1,1\})}(x) = x\,\frac{2^{n+3}\,n(n-1)(n-2)}{\prod_{1\le i<j\le n-1}(U_{j,n}-U_{i,n})}\,\det M_n^{\{1,1\}} + 16\,(n-1)(n-2)H_{n-2}(0),$$

where $M_n^{\{1,1\}}$ encodes classical zeros.

• A recent development exploits the correspondence between XHPs and points in George Wilson's adelic Grassmannian $Gr^{ad}$: every XHP family can be realized through generating functions given in terms of a Calogero–Moser pair $(X^\lambda, Z^\lambda)$, leading to closed determinantal expressions for each XHP,

$H_n^{(\lambda)}(x, y) = \frac{\det(\widetilde{X})}{s(n)} \sum_{k=0}^{N-1} \frac{n!}{(n-k)!} H_{n-k}(x, y) \cdot a \widetilde{X}^{-1} Z^{N-k-1} b + \text{(additional term if %%%%41%%%%)}.$

Here $\widetilde{X} = -X^\lambda + xI + 2yZ^\lambda$, and $a$, $b$ come from the explicit CM data (Paluso et al., 29 Jul 2025, Kasman et al., 2020).

6. Zero Distribution, Electrostatics, and Asymptotics

The zeros of XHPs divide into regular (real) zeros, which are distributed as in the classical Hermite case, and exceptional (complex) zeros, which are attracted to the zeros of the denominator Wronskian. For even partitions, after the scaling $x_k/\sqrt{2n}$, the real zeros of XHPs asymptotically follow the semicircle law: $$\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n - |\lambda|} f(x_{j,n}/\sqrt{2n}) = \frac{2}{\pi} \int_{-1}^1 f(x) \sqrt{1 - x^2}\, dx,$$ while the exceptional zeros converge at $O(1/\sqrt{n})$ to the zeros of the generalized Hermite polynomial $H_\lambda$ (Kuijlaars et al., 2014).

The energy function associated with the zeros demonstrates that the regular zeros minimize the logarithmic energy with respect to a modified weight, while the total set of zeros provides a saddle point (Horváth, 2016).

7. Connections to Quantum Systems, Integrability, and Physical Applications

Rational extensions of the harmonic oscillator constructed via XHPs yield exactly solvable quantum Hamiltonians with real spectra, which are either Hermitian or quasi/pseudo-Hermitian (in the non-Hermitian but real-spectrum case) (Midya, 2012, Marquette et al., 2012). The embedding of XHPs within superintegrable systems provides models for quantum systems exhibiting higher symmetries, shape-invariant potentials, and energy-dependent rational potentials that generalize the isotonic oscillator in both nonrelativistic and Dirac settings (Yeşiltaş et al., 2021).

The bispectral and integrable structure of XHPs is underscored by their generating functions being points in the adelic Grassmannian, with explicit connections to the KP hierarchy and Calogero–Moser flows (Kasman et al., 2020, Paluso et al., 29 Jul 2025). Through the bispectral anti-isomorphism, every exceptional Hermite family admits difference operators in degree, rich enough to support a variety of spectral and combinatorial phenomena (Gomez-Ullate et al., 2015).

8. Relations to Other Exceptional and Classical Families

There exist well-defined limit transitions and quadratic transformations that connect exceptional Hermite, Laguerre, and Jacobi families. For example, type III $X_m$-Hermite polynomials are obtained as asymptotic limits or via quadratic transformations of type III exceptional Laguerre or Jacobi polynomials,

$$\widehat{H}^{\mathrm{III},2m}_n(x) = \mathscr{L}^{\mathrm{III},m}_n(x^2; 1/2),$$

extending the classical $H_{2n}(x) = L_n^{-1/2}(x^2)$ paradigm (Quesne, 2023).

References to Key Results and Constructions

Result/Property	Reference(s)	Notes
Wronskian & Darboux structure	(Gomez-Ullate et al., 2013, Haese-Hill et al., 2015, Gomez-Ullate et al., 2020, Paluso et al., 29 Jul 2025)	General construction and completeness
Orthogonality/completeness	(Duran, 2013, Gomez-Ullate et al., 2019, Gomez-Ullate et al., 2020)	Conditions and full $L^2$-basis
Higher-order recurrence	(Durán, 2014, Gomez-Ullate et al., 2015, Grundland et al., 2022, Kasman et al., 2020)	Order $2\ell+3$ or minimal claimed
Ladder algebras, shape invariance	(Marquette et al., 2012, Marquette et al., 2012, Grundland et al., 2022)	PHAs and shape invariance structures
Determinantal formulas	(Simanek, 2022, Paluso et al., 29 Jul 2025)	Explicit matrix and CM representations
Zeros, semicircle law, asymptotics	(Kuijlaars et al., 2014, Horváth, 2016)	Real/complex zeros and electrostatics
Integrability, KP, CM, bispectrality	(Kasman et al., 2020, Paluso et al., 29 Jul 2025)	Grassmannian/generators/annihilators
Limit relations, classical links	(Quesne, 2023)	Quadratic/asymptotic transformations
Quantum systems, Dirac mapping	(Midya, 2012, Yeşiltaş et al., 2021)	Potentials, symmetry, energy dependence

Summary

Exceptional Hermite polynomials are higher-codimension, monodromy-free deformations of classical Hermite systems, indexed by partitions, and are characterized by their Wronskian structure, higher-order recurrence, and bispectral/differential-difference operator algebras. They furnish the building blocks for rational extensions of oscillator-like quantum systems, admit explicit determinantal and Calogero–Moser representations, and provide a nexus between spectral theory, integrable systems, and multivariate algebra. Their analytic, combinatorial, and operator-theoretic properties fundamentally extend the paradigm of classical orthogonal polynomials and facilitate the construction of exactly-solvable, and often superintegrable, physical models.