Supersymmetric Partner Potentials

• Supersymmetric partner potentials are defined as pairs of Hamiltonians connected via a superpotential, enabling systematic spectral mapping and exact solvability.
• They are constructed using algebraic factorization methods that yield nearly isospectral spectra and facilitate shape invariance and integrability.
• These potentials underpin applications in quantum control, spectral engineering, and non-Hermitian quantum systems by offering tunable energy states and controlled deformations.

Supersymmetric partner potentials are a central construct of one-dimensional supersymmetric quantum mechanics (SUSY QM), providing a systematic way to generate families of Hamiltonians with closely related spectra and underlying symmetry structures. They underpin exact solvability, hierarchical spectral design, and deep connections to integrability, special functions, and non-Hermitian spectral theory.

1. Algebraic Structure and Construction

Let $H_- = -d^2/dx^2 + V_-(x)$ be a one-dimensional Schrödinger operator. SUSY QM factorizes $H_-$ in terms of first-order operators built from a superpotential $W(x)$: $$A = \frac{d}{dx} + W(x),\quad A^\dagger = -\frac{d}{dx} + W(x).$$ The partner Hamiltonians are

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

with explicit partner potentials

$$V_-(x) = W^2(x) - W'(x),\quad V_+(x) = W^2(x) + W'(x).$$

The core property, verified directly, is that $A H_- = H_+ A$ and $A^\dagger H_+ = H_- A^\dagger$, intertwining the spectra of $H_-$ and $H_+$ (Ahmed et al., 2016, Ranjani et al., 2012, 2206.13020, Sekhon, 2022).

If the ground state $\psi_0^{(-)}(x)$ of $H_-$ is normalizable and nodeless, the superpotential can be chosen as $W(x) = -\psi_0'^{(-)}(x)/\psi_0^{(-)}(x)$. In this case, the spectrum of $H_+$ matches that of $H_-$ except the ground state of $H_-$, which is absent in $H_+$. Excited eigenstates are mapped by $A, A^\dagger$.

Generalizations include construction via nodeless half-bound states (non-normalizable zero-energy solutions), isospectral deformations, and higher-order intertwiners for more elaborate spectral control (Ahmed et al., 2016, Rosu et al., 2013).

2. Spectral Properties and Isospectrality

Supersymmetric partner potentials exhibit nearly isospectral spectra: if SUSY is unbroken (normalizable zero mode exists), the spectra coincide except for the ground state; if broken (no normalizable zero mode), they are exactly isospectral. Gozzi's criterion formalizes this at the wavefunction level: the difference in node count for same-energy eigenstates is one for unbroken SUSY and zero for broken SUSY (Ranjani et al., 2012).

Notably, applying the SUSY construction to nodeless but non-normalizable solutions (half-bound states) leads to partner potentials with strictly positive area in the sense of Simon's criterion, guaranteeing the absence of true bound states. Their negative counterparts always possess at least one bound state for arbitrary positive scaling (Ahmed et al., 2016).

Isospectral families can be generated by Darboux/Abraham-Moses transformations, introducing parametric deformation (e.g., quartic double-well potentials) without altering the spectrum (Rosu et al., 2013). Power-law or Liouville-reflectionless families, constructed with appropriate Bessel-function seeds, illustrate extension to continuous spectra or countably infinite level sets (Sukumar, 2021).

3. Explicit Examples and Model Potentials

SUSY partner construction has been applied to an array of canonical potentials:

• Harmonic Oscillator: Partner potentials arise as shifted or deformed oscillators, with closed-form superpotentials and explicit eigenfunction mappings. Isospectral deformations permit tunable ground state energies and physical applications such as fine control of Bose–Einstein condensation thresholds (Fellows et al., 2011).
• Morse Potential: The shape-invariant structure extends to all (real or even complex) parameter values, allowing for interpolation between bound and scattering spectral regimes and analytic continuation. The explicit superpotential $W(x) = |A| - e^{-x}$ ensures the construction is valid regardless of the sign of $A$ (Rath, 2015).
• Square Well and Box: Applying SUSY to box potentials leads to partner wells with trigonometric or rational singularities. These exhibit ladder spectra shifted by one energy level and underpin the quantum speed limit and control protocols due to their isospectrality (2206.13020, Sekhon, 2022, Koohrokhi, 2020).
• Delta and Multi-Delta Systems: Both first- and second-order SUSY transformations enable the manipulation of discrete and scattering spectra, with physical consequences for the sign and localization of delta-function defects (C. et al., 2010, Guilarte et al., 2014).

A selection of these examples (with superpotentials, partner potentials, and spectral features) is summarized in the following table:

Original Potential	Superpotential $W(x)$	Partner $V_+(x)$
$x^2$ (osc)	$x$	$x^2 + 2$
Morse	$|A| - e^{-x}$	$e^{-2x} - (2|A|-1)e^{-x} + |A|^2$
Box	$\frac{\pi}{L} \tan(\frac{\pi x}{L})$	$$\frac{\pi^2}{L^2}[\sec^2(\frac{\pi x}{L}) + \tan^2(\frac{\pi x}{L})]$$
Delta-well	See (C. et al., 2010) for full forms	Nontrivial, analytic, defect-localized

4. Shape Invariance and Integrability

Shape invariance is a crucial integrability criterion in SUSY QM: partner potentials retain their functional form under parameter transformation, up to an additive remainder. Explicitly, $V_+(x;a_0) = V_-(x;a_1) + R(a_0)$ with parameter map $a_1 = f(a_0)$. This property enables pure algebraic construction of spectra—the energy levels are sums of the additive shifts $R(a_k)$ over the parameter sequence.

Key classes of exactly solvable models—harmonic oscillator, Morse, Coulomb, radial oscillator—are shape invariant. Shape invariance underpins the algebraic solvability of the Wheeler–DeWitt equation in supersymmetric quantum cosmology and spectral design in higher-dimensional and curved space quantum systems (Jalalzadeh et al., 2022).

5. Extensions: Isospectral Deformation, Higher Order, and Non-Hermitian Cases

• Isospectral Deformation: By robustly varying the superpotential via integrable deformation schemes (Bernoulli or Ermakov–Pinney approaches), new partner families are generated with controlled spectral properties. This formalism yields, for example, one-parameter asymmetric double well potentials and systems with nontrivial localization of the zero-mode amplitude (anomalous localization region) (Rosu et al., 2013, Rosas-Ortiz et al., 2015).
• Higher-Order SUSY: Second or higher-order intertwining operators produce polynomial (nonlinear) supersymmetry, allowing for finer spectral engineering, including insertion or deletion of multiple discrete levels, isospectral phase-shifting for scattering states, and more complex algebraic structures such as polynomial Heisenberg algebras (C. et al., 2010, C et al., 2013).
• Complex and Non-Hermitian Partners: SUSY schemes extend into the complex domain with PT-symmetric, non-Hermitian partner potentials that nonetheless maintain real spectra in the unbroken PT-symmetric phase. Construction via Ermakov-based complex superpotentials or anti-PT inner products yields families with balanced gain/loss, band structures, and physical applications in optics and nuclear physics (Koohrokhi, 2020, Rosas-Ortiz et al., 2015, Koohrokhi et al., 2021).

6. Physical Applications and Contemporary Directions

Supersymmetric partner potentials underpin a diverse array of applications:

• Quantum Control: The intertwining relations facilitate transferable control protocols—such as shortcuts to adiabaticity and quantum speed limits—across the entire SUSY hierarchy, due to identical spectral (except ground state) and dynamical properties (2206.13020).
• Nonlinear Problems: Via mapping techniques (e.g., Lie symmetry), SUSY-derived potentials serve as exactly solvable backgrounds for inhomogeneous nonlinear Schrödinger equations, enabling closed-form construction of soliton and elliptic solutions in inhomogeneous media (C. et al., 22 Nov 2025).
• Cosmology and Quantum Gravity: Shape-invariant SUSY partner potentials appear as effective potentials in minisuperspace quantum cosmology, yielding exact spectra and wavefunctions for solvable Wheeler–DeWitt models (Jalalzadeh et al., 2022).
• Spectral Engineering: The machinery allows targeted insertion/removal of bound states, tailored scattering phase shifts, and construction of potentials with predetermined reflectionless or bandgap properties (Sukumar, 2021).

Supersymmetric quantum mechanics—with its partner potentials, algebraic underpinnings, and shape-invariance—continues to serve as a fundamental analytic framework in quantum theory, mathematical physics, and condensed matter, with extensions to integrable systems, non-Hermitian quantum mechanics, and quantum information (Sekhon, 2022, Ranjani et al., 2012).