Swanson Hamiltonian Overview

• Swanson Hamiltonian is a non-Hermitian quadratic oscillator defined by bosonic creation and annihilation operators with pair creation/annihilation terms.
• Its pseudo-Hermitian nature allows a similarity transformation to an equivalent Hermitian model, enabling exact diagonalization and detailed spectral analysis.
• Extensions include supersymmetric generalizations, noncommutative deformations, and time-dependent studies that broaden its impact in quantum mechanics.

Searching arXiv for recent and foundational papers on the Swanson Hamiltonian to ground the article in the literature. The Swanson Hamiltonian is a non-Hermitian quadratic oscillator Hamiltonian built from bosonic creation and annihilation operators and characterized by number-nonconserving quadratic terms. In its standard form,

$$H_S=\omega\,a^\dagger a+\alpha\,a^2+\beta\,(a^\dagger)^2$$

or, with the zero-point term retained,

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$

with $[a,a^\dagger]=1$ and typically real parameters $\omega,\alpha,\beta$ (Bagchi et al., 2015). For $\alpha\neq\beta$, the Hamiltonian is non-Hitian because the anomalous terms $a^2$ and $(a^\dagger)^2$ are not adjoints of each other with equal coefficients, yet it remains a central example of $\mathcal{PT}$-symmetric and pseudo-Hermitian quantum mechanics. Its importance derives from three connected facts: it is exactly diagonalizable in broad parameter regimes, it admits an equivalent Hermitian representation under a similarity transformation, and it has become a template for studying exceptional points, pseudo-bosonic structures, supersymmetric extensions, noncommutative deformations, time-dependent metrics, and multi-mode generalizations (Sinha et al., 2024).

1. Definition and operator structure

The defining feature of the Swanson Hamiltonian is the coexistence of the standard oscillator term with pair-annihilation and pair-creation terms. In the operator language used throughout the literature,

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$

where $\omega$ is the oscillator frequency and $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$0 control the anomalous quadratic couplings (Bagchi et al., 2015). The term $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$1 is the usual harmonic-oscillator Hamiltonian, while $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$2 and $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$3 are pair-annihilation and pair-creation contributions (Bagchi et al., 2015).

For real parameters, Hermiticity is lost whenever $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$4, since

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$5

and therefore $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$6 unless the two anomalous coefficients match in the appropriate way (Sinha et al., 2024). Nevertheless, the model is widely treated as $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$7-symmetric in the standard bosonic realization with parity acting as $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$8, $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$9, and time reversal sending $[a,a^\dagger]=1$0 (Sinha et al., 2024). In more general complex-parameter treatments, the $[a,a^\dagger]=1$1-symmetry condition is stated as $[a,a^\dagger]=1$2 (AlMasri et al., 2 Jun 2026).

This elementary-looking quadratic form is sufficiently flexible to generate several qualitatively different physical regimes. Depending on the signs of the effective mass parameter and of the squared effective frequency, the model can represent an ordinary harmonic oscillator, a negative-mass oscillator, an inverted oscillator or parabolic barrier, and a negative-mass parabolic barrier (Fernández et al., 2021). This classification is one reason the Swanson Hamiltonian functions as a canonical laboratory for non-Hermitian spectral theory.

2. Pseudo-Hermiticity and Hermitian equivalence

A basic structural result is that the Swanson Hamiltonian is pseudo-Hermitian. One introduces a positive-definite metric operator $[a,a^\dagger]=1$3 such that

$[a,a^\dagger]=1$4

and then defines the equivalent Hermitian Hamiltonian

$[a,a^\dagger]=1$5

which is manifestly Hermitian in the reference Hilbert space (Bagchi et al., 2015). This is the standard quasi-Hermitian construction for the model.

In Bogoliubov or Dyson form, one seeks a squeeze operator

$[a,a^\dagger]=1$6

that implements

$[a,a^\dagger]=1$7

with a corresponding expression for $[a,a^\dagger]=1$8, so that the transformed Hamiltonian becomes

$[a,a^\dagger]=1$9

The off-diagonal terms cancel when

$\omega,\alpha,\beta$0

and the reality of the transformed oscillator frequency requires

$\omega,\alpha,\beta$1

(Sinha et al., 2024).

In coordinate space, the equivalent Hermitian operator can also be written as a Schrödinger Hamiltonian. One explicit realization uses

$\omega,\alpha,\beta$2

which maps $\omega,\alpha,\beta$3 to

$\omega,\alpha,\beta$4

where

$\omega,\alpha,\beta$5

(Ramírez et al., 2022). This form makes explicit that the Swanson model can be interpreted as a harmonic or inverted oscillator depending on the sign of $\omega,\alpha,\beta$6.

The positivity of the metric is central. In one formulation,

$\omega,\alpha,\beta$7

is positive-definite for $\omega,\alpha,\beta$8, equivalently in the real-spectrum regime (AlMasri et al., 2 Jun 2026). This metric underlies the biorthogonal inner product and the associated physical interpretation of observables.

3. Spectrum, phases, and exceptional points

The spectral behavior of the Swanson Hamiltonian is controlled by the effective frequency

$\omega,\alpha,\beta$9

When $\alpha\neq\beta$0, the model has a real discrete spectrum

$\alpha\neq\beta$1

in the Hermitian representation, and therefore the original non-Hermitian Hamiltonian shares the same discrete spectrum in the unbroken regime (Sinha et al., 2024).

A more refined parameter-space classification uses

$\alpha\neq\beta$2

With these quantities, four regions arise in the $\alpha\neq\beta$3-plane. Region I, with $\alpha\neq\beta$4 and $\alpha\neq\beta$5, is an ordinary harmonic oscillator with discrete real spectrum

$\alpha\neq\beta$6

Region III, with $\alpha\neq\beta$7 and $\alpha\neq\beta$8, is a negative effective mass oscillator with discrete real spectrum of opposite sign,

$\alpha\neq\beta$9

Regions II and IV, where $a^2$0, correspond to parabolic-barrier regimes and support resonant ladders together with a real continuous band (Fernández et al., 2021).

In the broken-$a^2$1 treatment based on complex scaling, the same transition is described by taking $a^2$2 when $a^2$3. Then the discrete energies are written as

$a^2$4

so the spectrum appears in complex-conjugate form (Fernández et al., 2024). This formulation emphasizes resonance behavior and non-unitary dynamics.

Exceptional points occur on the critical hypersurface

$a^2$5

At this boundary, the effective frequency vanishes and the discrete levels collapse (Fernández et al., 2021). One description states that all discrete levels $a^2$6 coalesce at $a^2$7, and the Hamiltonian develops an infinite-order Jordan block at that energy (Fernández et al., 2021). A separate formulation identifies the same locus as the transition where the squeezing angle diverges, the similarity map becomes singular, and the Hamiltonian ceases to be diagonalizable (Sinha et al., 2024). These are not contradictory descriptions; they reflect different operator realizations and spectral embeddings.

One subtle boundary is $a^2$8. In that limit the effective mass diverges in the coordinate representation, and the Hamiltonian reduces, after a different similarity transformation, to

$a^2$9

with discrete generalized eigenfunctions $(a^\dagger)^2$0 and $(a^\dagger)^2$1 carrying positive and negative ladders of real eigenvalues (Fernández et al., 2021). This boundary is distinguished by anti-pseudo-Hermiticity rather than by exceptional points.

4. Biorthogonal, rigged-Hilbert-space, and complex-scaling formulations

Because the Swanson Hamiltonian is non-Hermitian, the conventional Hilbert-space eigenbasis is not always sufficient. In the non-$(a^\dagger)^2$2-symmetric phase, the eigenfunctions are treated as generalized eigenfunctionals in a Gel'fand triplet

$(a^\dagger)^2$3

with right and left eigenfunctionals satisfying

$(a^\dagger)^2$4

(Fernández et al., 2021). A similarity map $(a^\dagger)^2$5 sends the problem to an oscillator-type Hermitian operator, after which one constructs either Hermite-function states or parabolic-cylinder-function states and then transforms them back (Fernández et al., 2021).

In this setting, biorthogonality is imposed through a positive operator $(a^\dagger)^2$6, and the right and left generalized eigenfunctions provide completeness in the distributional sense (Fernández et al., 2021). This is particularly useful in the parabolic-barrier regime, where resonant states and continuous-spectrum states coexist.

An alternative formulation uses the Complex Scaling Method. One introduces the non-unitary dilatation

$(a^\dagger)^2$7

which transforms the coordinate-space Hamiltonian into a complex-scaled operator $(a^\dagger)^2$8 (Fernández et al., 2024). Further Bogoliubov-type maps reduce $(a^\dagger)^2$9 to a rotated harmonic or inverted oscillator $\mathcal{PT}$0 (Fernández et al., 2024).

The complex-scaled Hamiltonian admits right and left eigenfunctions $\mathcal{PT}$1 and $\mathcal{PT}$2 satisfying

$\mathcal{PT}$3

together with biorthonormality and completeness relations (Fernández et al., 2024). A central advantage of the complex-scaled representation is that it yields square-integrable eigenfunctions for suitable $\mathcal{PT}$4-sectors, whereas the rigged-Hilbert-space approach works directly with distributional Gamow-type states (Fernández et al., 2024). Both approaches are stated to reproduce the same physical decay rates and scattering quantities once the scaling sector is chosen appropriately (Fernández et al., 2024).

A related but distinct development is the fully pseudo-bosonic reformulation, where one replaces the canonical pair $\mathcal{PT}$5 by a $\mathcal{PT}$6-pseudo-bosonic pair $\mathcal{PT}$7 satisfying $\mathcal{PT}$8 on a common dense invariant domain. In that setting the Hamiltonian

$\mathcal{PT}$9

is diagonalized by a Bogoliubov-type transformation to

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$0

with a complete biorthogonal eigensystem and associated bi-coherent states (Bagarello, 2022). This formulation generalizes the Swanson construction beyond the standard adjoint relation $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$1.

5. Supersymmetry, pseudo-supersymmetry, and generalized differential realizations

A substantial branch of the literature embeds the Swanson Hamiltonian into supersymmetric and pseudo-supersymmetric frameworks. One starting point is the generalized differential realization

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$2

leading to the generalized Swanson Hamiltonian

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$3

(Midya et al., 2011). When $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$4 constant, the Hermitian equivalent can be transformed locally into a harmonic-oscillator equation, but the model need not be globally isospectral to the harmonic oscillator because the coordinate map $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$5 may cover a finite interval rather than the full real line (Midya et al., 2011). This non-isospectrality is one of the recurring subtleties of generalized Swanson systems.

Pseudo-supersymmetric factorization begins from a first-order differential operator $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$6 and the non-Hermitian Hamiltonian

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$7

A similarity transformation $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$8 yields a Hermitian Hamiltonian factorized as $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha a^2+\beta(a^\dagger)^2,$$9, with partner $\omega$0, intertwining operator $\omega$1, and pseudo-supersymmetric relation $\omega$2 (Yeşiltaş, 2011). For a specific choice of $\omega$3, $\omega$4, and $\omega$5, this construction produces isotonic and nonlinear isotonic oscillator potentials $\omega$6 and $\omega$7, along with exact eigenvalues and confluent-hypergeometric or Laguerre eigenfunctions (Yeşiltaş, 2011).

The second-derivative pseudo-supersymmetric generalization pushes the factorization further. One defines a diagonal SSUSY Hamiltonian

$\omega$8

together with second-order differential operators

$\omega$9

where $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$00 and $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$01 (Bagchi et al., 2015). These satisfy the intertwining relations

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$02

The associated quasi-Hamiltonian

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$03

is a fourth-order differential operator (Bagchi et al., 2015). For the special case $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$04, it factorizes as

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$05

subject to the perfect-square constraint

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$06

(Bagchi et al., 2015). This construction explicitly embeds the non-Hermitian Swanson oscillator into a higher-order supersymmetric algebra.

A related one-step extension uses a pseudo-Hermite-polynomial seed $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$07, $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$08, to generate a rationally extended Hermitian partner $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$09, and then pulls this back by similarity to a non-Hermitian extended Swanson Hamiltonian $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$10 (Bagchi et al., 2015). The extended model has higher-order ladder operators $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$11, $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$12, a polynomial Heisenberg algebra, and a spectrum

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$13

which differs from the equally spaced standard Swanson spectrum by the presence of an extra level (Bagchi et al., 2015).

6. Extensions, deformations, and dynamical developments

The Swanson Hamiltonian has served as the seed for numerous extensions. In two dimensions, one construction defines

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$14

which, after similarity transformation, becomes the sum of two extended one-dimensional oscillators. Under the resonance condition

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$15

one obtains algebraically independent integrals of motion proving superintegrability (Bagchi et al., 2015).

A different two-dimensional generalization is formulated on a noncommutative plane. There the Hamiltonian $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$16 is diagonalized by pairs of $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$17-pseudo-bosons $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$18, leading to explicit biorthogonal eigenstates, eigenvalues

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$19

and bi-coherent states with a resolution of the identity on a dense domain (Bagarello et al., 2017). This indicates that the algebraic Swanson mechanism persists under simultaneous non-Hermiticity and spatial noncommutativity.

Time dependence introduces an additional layer of structure. For the generalized time-dependent Hamiltonian

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$20

one seeks a time-dependent Dyson map $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$21 such that

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$22

is Hermitian, equivalently

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$23

(Fring et al., 2016). With an $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$24-type ansatz for $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$25, the Hermiticity of $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$26 reduces to coupled nonlinear differential equations for time-dependent parameters, after which the Lewis–Riesenfeld invariant method yields exact solutions (Fring et al., 2016).

The dynamical behavior of the Swanson oscillator has also been analyzed semiclassically. For the $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$27-symmetric quadratic Hamiltonian

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$28

metriplectic flow equations produce closed-form expressions for the phase-space metric and trajectories (Graefe et al., 2014). Although the spectrum remains real and discrete after similarity transformation to a harmonic oscillator with frequency $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$29, the classical metric, the real phase-space trajectories, and Gaussian wave packets can diverge in finite time for $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$30 (Graefe et al., 2014). This is frequently cited as a caution against identifying real spectrum with completely benign dynamics.

Finite-temperature pseudo-Hermitian thermodynamics has likewise been worked out. In the $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$31-unbroken regime, the partition function of the equivalent harmonic oscillator is

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$32

while in the $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$33-broken inverted-oscillator regime one obtains a resonance-sector partition function

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$34

with real internal energy

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$35

(Ramírez et al., 2022). The same framework supports double-time Green’s functions and linear response under periodic perturbations (Ramírez et al., 2022).

More recently, out-of-time-ordered correlators have been computed for the exactly solvable quadratic Swanson Hamiltonian and for Kerr and driven extensions. In the unbroken phase,

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$36

while in the broken phase with $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$37,

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$38

The exponential growth in the broken phase is explicitly interpreted as linear instability rather than quantum chaos (AlMasri et al., 2 Jun 2026).

7. Generalizations to coupled and fermionic models

The Swanson construction has also been exported to coupled and fermionic settings. A two-mode interacting system is built from two Swanson-like quadratic Hamiltonians plus a $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$39-invariant bilinear interaction,

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$40

with the full problem reduced to a $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$41 non-Hermitian matrix (Bagchi et al., 2021). In the simplifying case $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$42, the eigenvalues are available in closed form, and exceptional points arise when the nested square roots collapse. Under the additional conditions

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$43

all four roots coalesce at $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$44, producing fourth-order exceptional points (Bagchi et al., 2021).

A fermionic extension with two fermionic oscillators exhibits exceptional points and a quantum phase transition. The model is described as a fermionic extension of the Swanson scheme with bilinear couplings that do not conserve particle number, and its exceptional points are characterized by coalescing eigenstates and self-orthogonality with respect to the biorthogonal inner product (Sinha et al., 2024). The same work emphasizes that the Swanson oscillator remains a standard benchmark for non-Hermitian quantum mechanics (Sinha et al., 2024).

A three-site fermionic Swanson-like model provides an exactly solvable example of a triple exceptional point. After reduction to a traceless $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$45 block

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$46

the secular equation is cubic, and the EP3 locus is given in closed form by

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$47

(Bagchi et al., 2 Jun 2026). The same model exhibits a non-empty “corridor of unitarity” near the EP3 when perturbations are scaled by fractional powers of a small parameter (Bagchi et al., 2 Jun 2026). This suggests a broader role for Swanson-type constructions in higher-order non-Hermitian degeneracies.

A final strand of generalization is interpretive rather than algebraic. The Swanson oscillator has been used as the non-Hermitian structure underlying a quantization of the Schwarzschild horizon area. In that framework, a similarity transformation maps the Swanson Hamiltonian to a scaled harmonic oscillator and yields the area spectrum

$$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$48

together with a logarithmic entropy correction proportional to $$H_S=\omega\Bigl(a^\dagger a+\tfrac12\Bigr)+\alpha\,a^2+\beta\,(a^\dagger)^2,$$49 (Bagchi et al., 2024). This does not alter the core quantum-mechanical definition of the model, but it illustrates the portability of the Swanson formalism across mathematical physics contexts.

The Swanson Hamiltonian therefore occupies a distinctive place in non-Hermitian quantum theory: it is simple enough to be solved exactly, but rich enough to support pseudo-Hermitian metrics, biorthogonal spectral theory, supersymmetric factorizations, exceptional-point phenomena of several orders, and a wide range of deformations and applications. Its enduring role in the literature reflects this combination of algebraic tractability and structural depth.