Quasi-Exactly Solvable Systems (QES)

• QES systems are differential operators whose finite algebraic sector yields a limited set of solvable spectral values, contrasting with fully solvable models.
• They leverage hidden Lie algebras, gauge rotations, and terminating polynomial recurrences to reveal invariant subspaces in diverse settings.
• Applications include one-dimensional sextic oscillators, PT-symmetric non-Hermitian systems, and many-body Calogero-Sutherland models with explicit zero-energy bound states.

QES most commonly denotes quasi-exactly solvable systems in spectral theory and mathematical physics: differential operators for which only a finite part of the spectral problem can be solved algebraically, in contrast with exactly solvable systems that preserve an infinite flag of invariant polynomial subspaces. In the literature represented here, QES includes one-dimensional sextic and double sinh-Gordon models, PT-symmetric non-Hermitian systems, rational many-body extensions of the truncated Calogero-Sutherland model, and higher-dimensional constructions on $S^n$. The same acronym is also used in other domains, notably for quantum extremal surfaces and qualified electronic signatures, so context is essential (Yadav et al., 2024, Jr. et al., 2014, Khodahami et al., 17 Jun 2025, Bicakci et al., 10 Jan 2026).

1. Definition and algebraic scope

The defining distinction between exact solvability and quasi-exact solvability is algebraic. For an exactly solvable operator, there exists an infinite flag of finite-dimensional invariant polynomial spaces,

$$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$

whereas for a QES operator there is only one finite-dimensional invariant subspace,

$H\mathcal P_k \subseteq \mathcal P_k,$

so only a finite portion of the spectrum and eigenfunctions is obtained algebraically (Jr. et al., 2014). In the rationally extended truncated Calogero-Sutherland construction, this distinction is made especially explicit: the system is QES because the zero-energy state and its wavefunction are obtained in closed form, while the full spectrum is not necessarily available (Yadav et al., 2024).

This definition covers several technically distinct mechanisms. In some models, the solvable sector is encoded by hidden Lie-algebra representations acting on polynomial modules. In others, it is encoded by a terminating recursion for Bender-Dunne polynomials whose zeros give the algebraic energies. The common feature is not full diagonalizability in closed form, but the existence of a controlled algebraic subsector. This suggests that QES is best understood as a property of an operator’s partially algebraic sector, rather than as a statement about the entire spectrum.

The same notion extends beyond Hermitian one-dimensional Schrödinger problems. The supplied literature includes QES systems on curved spaces, PT-symmetric non-Hermitian Hamiltonians, many-body radial reductions, and supersymmetric partner constructions. It also includes cases in which the physically distinguished state is not a low-lying level but a regular, square-integrable zero-energy state, which occupies the threshold between bound and continuum behavior (Yadav et al., 2024).

2. Hidden algebras and constructive machinery

A recurring theme is that QES models are generated by hidden algebraic structures, gauge rotations, and variable changes that expose an invariant finite-dimensional space. Representative constructions appear across several distinct settings (Yadav et al., 2024, Jr. et al., 2014, Contreras-Astorga et al., 2023, Khare et al., 2011, Mandal et al., 2019).

Setting	Algebraic structure or device	Resulting QES feature
Rationally extended truncated Calogero-Sutherland model	$so(2,1)$ potential algebra and point canonical transformation	Three potential families with the same algebraic energy relation
Sphere $S^n$	Hidden $gl_n$ realized by first-order differential operators on $RP^n$	Rational ES/QES potentials and finite-dimensional polynomial modules
Sextic oscillator	Hidden $\mathfrak{sl}_2(\mathbb R)$ in $\tau=x^2$	$(N+1)$ algebraic eigenstates
PT-symmetric double sinh-Gordon and related models	Bender-Dunne polynomial recursion	Finite algebraic sectors determined by truncation
Two-dimensional PT-symmetric nonlinear system	Canonical transformation to a sextic QES problem	First few QES levels from Bender-Dunne polynomials

In the extended truncated Calogero-Sutherland model, the $$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$0 generators are written as

$$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$1

with

$$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$2

The Casimir yields a Schrödinger-type equation,

$$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$3

and a point canonical transformation,

$$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$4

is then used to match the algebraic problem to the many-body radial equation (Yadav et al., 2024).

On $$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$5, the hidden algebra is $$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$6, realized by

$$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$7

acting on

$$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$8

The ES operator preserves an infinite flag, whereas the QES deformation adds the raising generators and preserves only one $$\mathcal P_0 \subset \mathcal P_1 \subset \cdots, \qquad H\mathcal P_k \subseteq \mathcal P_k,$$9 (Jr. et al., 2014).

For the sextic oscillator,

$H\mathcal P_k \subseteq \mathcal P_k,$0

the gauge-rotated operator in $H\mathcal P_k \subseteq \mathcal P_k,$1 closes on $H\mathcal P_k \subseteq \mathcal P_k,$2, again exposing a finite-dimensional polynomial module when $H\mathcal P_k \subseteq \mathcal P_k,$3 (Contreras-Astorga et al., 2023). In double sinh-Gordon and related PT-symmetric models, the algebraic sector is instead encoded in Bender-Dunne polynomials obeying a three-term recurrence, with quasi-exact solvability appearing when the recurrence truncates (Khare et al., 2011). The two-dimensional PT-symmetric nonlinear system reaches the same type of algebraic closure after a canonical transformation to a sextic effective Hamiltonian (Mandal et al., 2019).

3. Many-body rational QES potentials and zero-energy bound states

A recent many-body realization of QES is based on a rationally extended truncated Calogero-Sutherland model. The extended Hamiltonian is

$H\mathcal P_k \subseteq \mathcal P_k,$4

with

$H\mathcal P_k \subseteq \mathcal P_k,$5

and

$H\mathcal P_k \subseteq \mathcal P_k,$6

The model is “truncated” because interactions occur only among neighbors within a range $H\mathcal P_k \subseteq \mathcal P_k,$7, rather than all particle pairs. The rational term $H\mathcal P_k \subseteq \mathcal P_k,$8 makes the radial problem suitable for generating new QES rational potentials (Yadav et al., 2024).

Three admissible $H\mathcal P_k \subseteq \mathcal P_k,$9 realizations generate three distinct QES potential families, all supporting regular, normalizable $so(2,1)$0 states after appropriate parameter restrictions:

• Case I: $so(2,1)$1, with normalizability conditions

$so(2,1)$2

• Case II: $so(2,1)$3, with acceptable asymptotics requiring

$so(2,1)$4

• Case III: $so(2,1)$5, with convergence of the normalization integral when

$so(2,1)$6

The construction is notable because the three potentials share the same algebraic energy relation while differing in the functional form of $so(2,1)$7, $so(2,1)$8, and the induced $so(2,1)$9. The paper emphasizes that, unlike the earlier Bagchi–Quesne zero-energy QES construction where only one of the three algebraic classes yielded a normalizable $S^n$0 state, here all three classes produce normalizable zero-energy solutions once the coupling constants are restricted appropriately (Yadav et al., 2024).

The physical significance of these states lies in the role of $S^n$1 as a threshold. A zero-energy state can be bound or unbound depending on whether its wavefunction is normalizable. In this setting, the algebraic construction yields explicit square-integrable wavefunctions and therefore explicit examples of regular zero-energy bound states in an extended many-body system. Mathematically, the result shows that a nontrivial family of QES rational potentials can be generated from a many-body model through $S^n$2 algebra and point canonical transformation.

4. Geometric, PT-symmetric, and non-Hermitian realizations

QES systems also arise naturally on curved spaces. On the sphere $S^n$3, one introduces

$S^n$4

which maps the sphere to a simplex. In these variables the contravariant metric becomes polynomial,

$S^n$5

with determinant

$S^n$6

The ES potential is rational, and the QES deformation adds a raising-operator term to the gauged ES Hamiltonian. The resulting QES system is completely integrable for $S^n$7 and non-maximally superintegrable for $S^n$8, but there is no separable coordinate system in which it is exactly solvable (Jr. et al., 2014).

A second major branch consists of PT-symmetric non-Hermitian models. One standard example is

$S^n$9

with PT invariance under the generalized parity transformation

$gl_n$0

combined with time reversal. For each integer $gl_n$1, the first $gl_n$2 energy levels and eigenfunctions can be found exactly through the Bender-Dunne polynomial construction. The model exhibits a PT phase structure: for even $gl_n$3, all eigenvalues are complex for any $gl_n$4, while for odd $gl_n$5 the spectrum is real for $gl_n$6 and PT symmetry breaks spontaneously at $gl_n$7 (Mandal et al., 2013).

The double sinh-Gordon family shows that certain perturbations preserve quasi-exact solvability. Starting from Hermitian or PT-invariant complex double sinh-Gordon Hamiltonians, one may add terms such as

$gl_n$8

or the combined perturbation with an additional $gl_n$9-type contribution, and the resulting systems remain QES. Under the anti-isospectral transformation

$RP^n$0

the hyperbolic models become periodic trigonometric QES models with reversed, sign-flipped algebraic spectra (Khare et al., 2011).

A further example begins from the two-dimensional PT-symmetric nonlinear system

$RP^n$1

which is represented by a non-Hermitian Hamiltonian with position-dependent mass,

$RP^n$2

A canonical transformation,

$RP^n$3

maps the problem to a sextic QES system whose first few levels can be computed by the Bender-Dunne polynomial method (Mandal et al., 2019).

5. Supersymmetry, partner potentials, and algebraic persistence

Supersymmetric transformations provide a stringent test of how robust quasi-exact solvability is under spectral deformation. For the sextic oscillator

$RP^n$4

the hidden $RP^n$5 structure is visible in the variable $RP^n$6. The revisited SUSY analysis shows that this hidden algebra is inherited by the first-order SUSY partner potential $RP^n$7 only for $RP^n$8. For fixed $RP^n$9, the partner still has $\mathfrak{sl}_2(\mathbb R)$0 exact eigenpolynomial solutions, but the simple $\mathfrak{sl}_2(\mathbb R)$1-algebraic description in the same variable is lost (Contreras-Astorga et al., 2023).

The partner potential splits into a polynomial part and rational terms. Its polynomial component is given by the same sextic QES form but with a shifted non-integer parameter,

$\mathfrak{sl}_2(\mathbb R)$2

The exact SUSY partner states are odd-parity zero modes, and the partner potential can be represented as the sum of a polynomial and rational parts. A confluent second-order SUSY transformation is also implemented for a modified QES sextic potential possessing energy reflection symmetry (Contreras-Astorga et al., 2023).

The spectral analysis in that work treats $\mathfrak{sl}_2(\mathbb R)$3 as a continuous real parameter in

$\mathfrak{sl}_2(\mathbb R)$4

and computes highly accurate values of $\mathfrak{sl}_2(\mathbb R)$5 for the three lowest states $\mathfrak{sl}_2(\mathbb R)$6. The critical value

$\mathfrak{sl}_2(\mathbb R)$7

marks the onset above which tunneling effects can occur. This suggests that quasi-exact solvability can coexist with nonperturbative phenomena, but the hidden algebra need not survive unchanged under SUSY transformations.

6. Other established meanings of “QES”

Although quasi-exact solvability is the dominant meaning in the mathematical-physics literature summarized above, the acronym is not unique. In other fields represented here, “QES” denotes several unrelated concepts (Khodahami et al., 17 Jun 2025, Pedraza et al., 2021, Bicakci et al., 10 Jan 2026, Koch et al., 15 Dec 2025, Dong et al., 2021, Nguyen et al., 2021, Xu et al., 3 Feb 2026).

Usage	Definition in the cited work	Representative paper
Quantum extremal surface	Codimension-2 spacelike surface extremizing generalized entropy in gravity	(Khodahami et al., 17 Jun 2025, Pedraza et al., 2021)
Qualified electronic signature	High-assurance e-signature under eIDAS, often anchored in QTSP and QSCD infrastructure	(Bicakci et al., 10 Jan 2026, Koch et al., 15 Dec 2025)
Quantum unitary evolution score	Benchmark for Hamiltonian simulation based on ancilla success probability	(Dong et al., 2021)
Quantum Embedding Search	Automated search for quantum embedding architectures in QML	(Nguyen et al., 2021)
Quantized Evolution Strategies	Backpropagation-free fine-tuning of quantized LLMs in discrete weight space	(Xu et al., 3 Feb 2026)

In gravity, a QES is the surface $\mathfrak{sl}_2(\mathbb R)$8 that extremizes

$\mathfrak{sl}_2(\mathbb R)$9

and recent work has even proposed a revised prescription in which entropy is obtained from a weighted sum over multiple candidate surfaces rather than by minimizing a single generalized entropy functional (Khodahami et al., 17 Jun 2025). In digital identity and security engineering, QES denotes qualified electronic signature, including architectures that bind a virtual FIDO2 authenticator to QES-grade PKCS#11 hardware or distribute QES creation through privacy-preserving collaborative computations (Bicakci et al., 10 Jan 2026, Koch et al., 15 Dec 2025).

The acronym therefore requires domain-specific disambiguation. Within mathematical physics, however, QES remains firmly associated with the partially algebraic spectral theory of Schrödinger-type operators, hidden Lie algebras, polynomial invariants, and finite exactly computable sectors.