SUSY Partner Hamiltonians

• SUSY Partner Hamiltonians are quantum systems defined by factorizing a Hamiltonian using a superpotential, leading to nearly isospectral partners with mapped eigenfunctions.
• They exhibit rich algebraic structures, including polynomial Heisenberg and Lie algebras, which facilitate the construction of ladder operators and symmetry analysis.
• Applications include spectral engineering, scattering theory, and integrable systems, offering analytic and numerical methods for potential design and control.

Supersymmetric (SUSY) partner Hamiltonians are pairs or families of quantum mechanical Hamiltonians constructed via supersymmetric quantum mechanics such that their spectra and eigenfunctions are related by intertwining differential operators. In one-dimensional, matrix, and position-dependent-mass settings, the SUSY formalism allows for the algorithmic generation of exactly or quasi-exactly solvable potentials, spectral engineering, and analytic control over spectral modifications. SUSY partners appear throughout quantum mechanics, mathematical physics, and mathematical approaches to integrable systems, including connections to special function theory, non-Hermitian systems, and quantum field theory. Modern results extend the construction to non-Hermitian, position-dependent-mass, and multi-channel Hamiltonians, and reveal deep links to algebraic structures such as Lie algebras, polynomial Heisenberg algebras, and relations to Painlevé transcendents.

1. Basic Theory of SUSY Partner Hamiltonians

The standard formalism arises from factorizing a given Hamiltonian $H_-$ using a first-order differential operator constructed from a "superpotential" $W(x)$. For a Schrödinger operator,

$$H_- = -\frac{d^2}{dx^2} + V_-(x) = A^\dagger A + \epsilon,$$

with

$$A = \frac{d}{dx} + W(x), \qquad A^\dagger = -\frac{d}{dx} + W(x),$$

the SUSY partner Hamiltonian is

$$H_+ = A A^\dagger + \epsilon = -\frac{d^2}{dx^2} + V_+(x),$$

where

$V_\pm(x) = W^2(x) \mp W'(x) + \epsilon.$

The intertwining relations

$$A H_- = H_+ A, \qquad A^\dagger H_+ = H_- A^\dagger$$

guarantee that (except possibly for a discrete "missing" state) the spectra of $H_-$ and $H_+$ coincide, and their eigenfunctions are mappped bijectively by the intertwiners. This construction generalizes to higher-order and matrix-valued operators, as well as to cases with position-dependent mass or non-Hermitian settings (Williams et al., 2017Socorro et al., 2019Bagarello, 2020Ioffe et al., 2016).

In the context of matrix Hamiltonians, including the Jaynes–Cummings model and the spherically symmetric Pauli Hamiltonian, the SUSY structure is encoded via block-diagonal Hamiltonians and nilpotent supercharges (Ateş et al., 28 Apr 2025Junker, 11 Apr 2025). For position-dependent-mass systems and multi-channel scattering, the factorization is extended by accounting for variable mass and matrix intertwiners (Delgado et al., 2019Pupasov et al., 2010).

2. Classes and Algebraic Structures of SUSY Partners

SUSY partners manifest in several algebraic forms that classify the ladder and symmetry structures of the systems:

a. Polynomial Heisenberg Algebras: For Hamiltonians such as the SUSY partners of the harmonic oscillator, the natural ladder operators close a polynomial deformation of the Heisenberg algebra, often of order determined by the SUSY step (1205.62391402.5926). Concretely, for a $k$-th order SUSY transformation, the ladder operators $L_k^\pm$ close

$$[H_k, L_k^\pm] = \pm L_k^\pm, \quad [L_k^-, L_k^+] = P_{2k}(H_k)$$

with a structure polynomial $P_{2k}$.

b. Lie Algebraic Structures: For quasi-exactly solvable (QES) models, the SUSY partners possess hidden algebraic symmetry, often $\mathfrak{sl}_2(\mathbb{R})$, realized at special values of parameters (Contreras-Astorga et al., 2023). These structures control the solvable part of the spectrum and their breakdown in general for SUSY partners modifies the analytic sector.

c. Coupled SUSY: More sophisticated algebraic generalizations involve pairs of operators ($a,a^\dagger$, $b,b^\dagger$) whose factorization relations differ by spectral shifts, yielding $\mathfrak{su}(1,1)$ ladder algebras and enabling coverage of the full spectrum via raising and lowering operations (Williams et al., 2017).

d. Shape Invariance and Isospectrality: Partner potentials are termed "shape-invariant" if their forms remain unchanged up to parameter shifts (e.g. $\lambda\to\lambda-1$ in Rosen–Morse II), yielding exact analytic solution formulas and closed algebraic expressions for the energy spectrum (Millán et al., 2023Junker, 11 Apr 2025).

3. Construction and Properties of SUSY Partners

A central element is the choice of "seed solution" $u(x)$, which leads to different types of spectral modifications:

• Ground-state (typical SUSY): Choosing the normalized ground state as the seed removes the lowest level from the original spectrum; higher SUSY chains systematically strip additional levels.
• Virtual/Anti-bound Seed: Using nonnormalizable solutions (e.g., anti-bound states) can create new bound states in the SUSY partner, "lifting" anti-bound poles to physical levels (Millán et al., 2023).
• Redundant/Quasi-bound Seed: Using solutions associated with "redundant" S-matrix poles yields strictly isospectral rational extensions: the spectrum is unchanged but the partner potential differs nontrivially.

For matrix, multidimensional, or operator-valued seeds (e.g., in the Jaynes–Cummings or Dirac frameworks), intertwiners act as matrix operators, and the resulting hierarchies can connect families of Hamiltonians distinguished by parameter shifts (detuning, e.g.) (Ateş et al., 28 Apr 2025Bagchi et al., 2021).

Table: Types of SUSY transformations and their spectral effects (Rosen–Morse II)

Seed choice	Spectral effect in $H_+$	Partner potential
Ground state	Lowest bound state deleted	Shape-invariant shift
Anti-bound (virtual)	New bound state added	Rational extension
Redundant pole	Isospectral (no change), state moves	Rational extension

4. Applications: Spectral Engineering, Dynamics, and Special Function Theory

SUSY partner Hamiltonians are used for diverse analytic and computational applications:

• Spectral Design and Engineering: SUSY transformations permit systematic adjustment of the bound spectrum, gaps, or resonant structure of Hamiltonians, applicable in quantum optics (e.g., Jaynes–Cummings hierarchies) (Ateş et al., 14 Dec 2025Ateş et al., 28 Apr 2025).
• Scattering Theory and S-Matrix Poles: SUSY modifies the pole structure of the S-matrix in single- and coupled-channel systems, controllably removing, adding, or reshuffling bound, anti-bound, and redundant poles corresponding to physical and unphysical states (Millán et al., 2023Pupasov et al., 2010). In the two-channel case, the "eigenphase preserving" SUSY transformations enable decoupling eigenphase and mixing angle fitting, critical for inverse scattering problems.
• Connection to Special Functions and Integrable Systems: The algebraic structures and ladder operators for SUSY partners underlie connections to Painlevé transcendents (e.g., Painlevé IV/V), quasi-exactly solvable models, and the solution of non-linear ODEs (C et al., 2016Bermudez et al., 2014). Explicit construction of coherent states, their overcomplete resolutions of the identity, and analytical expressions for geometric phases are achieved for SUSY partners (1402.59261205.6239).
• Non-Hermitian Quantum Mechanics: The SUSY framework generalizes to non-Hermitian cases using two independent complex superpotentials. The intertwining ensures the reality (or quasi-reality) of the spectrum, biorthogonality, and Gazeau–Klauder–type bi-coherent states (Bagarello, 2020).
• Position-Dependent Mass and Higher Dimensions: For position-dependent mass systems, adapted intertwiners and mass-dependent superpotentials provide analytic solutions to variable-mass quantum systems, relevant in condensed matter contexts (Delgado et al., 2019Ioffe et al., 2016).
• Riemann Zeta and XP Models: SUSY–QM constructions lead to Hamiltonians whose spectral properties encode zeros of special functions, such as the Riemann zeta-function along the critical line (García-Muñoz et al., 2023).

5. Hierarchies, Shape Invariance, and SUSY Ladders

SUSY transformations can be iterated to build hierarchies of Hamiltonians (e.g., for the Jaynes–Cummings model, each SUSY step shifts the detuning and deletes/adds two levels). In shape-invariant models, such as the Rosen–Morse II and radial harmonic oscillator or Coulomb systems, the entire spectrum can be constructed recursively via parameter shifts and algebraic formulas (Millán et al., 2023Junker, 11 Apr 2025Ateş et al., 28 Apr 2025).

For QES and truncated oscillator systems, first- and second-order SUSY transformations yield sequences of Hamiltonians whose algebraic sectors are fully controlled, with explicit analytic expressions for eigenvalues and eigenfunctions, and clear algebraic decomposition of their spectrum (Contreras-Astorga et al., 2023C et al., 2016).

6. Advanced Topics: Coupled Supersymmetry and Non-Hermitian Extensions

Coupled SUSY generalizes the standard two-partner construction to a system of four Hamiltonians, related by pairs of intertwiners whose products differ by constant shifts. This structure enables ladder construction over the full spectrum (including multi-step interleaved eigenvalues) and closes on nontrivial Lie algebras such as $\mathfrak{su}(1,1)$, supporting families of generalized coherent states and uncertainty relations beyond the canonical case (Williams et al., 2017).

Non-Hermitian SUSY approaches relax the Hermiticity restriction on the partner Hamiltonians, allowing two independent complex superpotentials. This leads to biorthogonality, spectra that can nevertheless be real or pseudo-real, and construction of bi-coherent states with complete sets and temporal stability (Bagarello, 2020). Applications include real-world models such as Black-Scholes.

7. Numerical and Analytical Methods for SUSY Partners

The Rayleigh–Ritz variational method and Riccati–Padé techniques have demonstrated high-precision extraction of spectra for analytically intractable SUSY partners, including non-analytic or cusp-like superpotentials. The separation of parity and careful basis selection yields exponential convergence and high accuracy, essential for benchmarking and spectral design applications (Fernández, 2012).

The Lagrange–mesh method enables arbitrary-precision calculation of energy levels and eigenstates in QES and multi-well settings, crucial for verifying analytic predictions and detailed study of tunneling/inversion phenomena in the context of SUSY partners (Contreras-Astorga et al., 2023).

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References:

• "Supersymmetric Quantum Mechanics: two factorization schemes, and quasi-exactly solvable potentials" (Socorro et al., 2019)
• "Coupled Supersymmetry and Ladder Structures Beyond the Harmonic Oscillator" (Williams et al., 2017)
• "Harmonic Oscillator SUSY Partners and Evolution Loops" (C, 2012)
• "The SUSY partners of the QES sextic potential revisited" (Contreras-Astorga et al., 2023)
• "SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations" (C et al., 2016)
• "Painlevé IV Coherent States" (Bermudez et al., 2014)
• "Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics" (Fernández, 2012)
• "SUSY partners and $S$-matrix poles of the one dimensional Rosen-Morse II Hamiltonian" (Millán et al., 2023)
• "SUSY Method for the Three-Dimensional Schrödinger Equation with Effective Mass" (Ioffe et al., 2016)
• "Ladder operators for the Ben Daniel-Duke Hamiltonians and their SUSY partners" (Delgado et al., 2019)
• "Dirac Hamiltonian in a supersymmetric framework" (Bagchi et al., 2021)
• "On the SUSY structure of spherically symmetric Pauli Hamiltonians" (Junker, 11 Apr 2025)
• "SUSY hierarchies of Jaynes-Cummings Hamiltonians with different detuning parameters" (Ateş et al., 28 Apr 2025)
• "Expected values for SUSY hierarchies of Jaynes-Cummings Hamiltonian" (Ateş et al., 14 Dec 2025)
• "Susy for non-Hermitian Hamiltonians, with a view to coherent states" (Bagarello, 2020)
• "Eigenphase preserving two-channel SUSY transformations" (Pupasov et al., 2010)
• "High-Order SUSY-QM, the Quantum XP Model and zeroes of the Riemann Zeta function" (García-Muñoz et al., 2023)