Deformed Supersymmetric Quantum Mechanics

• Deformed Supersymmetric Quantum Mechanics is a framework where standard SUSY QM is modified through algebraic, geometric, and analytic deformations to regularize singularities and extend solvability.
• The methodology employs adjustments to superalgebras, modified potential and kinetic terms, and techniques like planarization and deformed shape invariance to restore isospectrality.
• These advances yield richer multiplet structures and connect to non-Hermitian, noncommutative, and discrete systems, enabling precise spectral and integrability analyses.

Deformed supersymmetric quantum mechanics (DSQM) encompasses a broad collection of models in which the conventional supersymmetric quantum mechanical structures are systematically altered by algebraic, analytic, or geometric deformations. These deformations may introduce parameter-dependent modifications to superalgebras, kinetic and potential terms, path integration contours, or underlying geometric backgrounds. Such approaches play a critical role in regularizing singularities, connecting non-Hermitian and noncommutative settings, constructing generalized multiplets, and facilitating exact analysis using integrable systems frameworks.

1. Algebraic and Geometric Deformation Principles

DSQM can involve several types of deformations, classified as follows:

• Algebraic Deformations: Modifying the representation or closure relations of the superalgebra via parameters that deform commutators/anticommutators or the structure of supercharges. Examples include $q$-deformation (with nontrivial scaling operators) (Gavrilik et al., 2013), polynomial superalgebras in higher-derivative or parasupersymmetric models (Spiridonov, 16 Apr 2024), and deformations arising from centrally extended supergroups such as $SU(2|1)$ or $SU(2|2)$ (Ivanov et al., 2013, Sidorov, 2019).
• Complex and Topological Deformations: Extending the domain of the coordinate or configuration space into the complex plane, often via multisheeted ("tobogganic") contours, to regularize singularities or accommodate nontrivial topology (Znojil, 2011).
• Geometric and Metric Deformations: Introducing position-dependent mass, varying spatial envelopes, or noncommutative geometry. Notable are position-dependent effective mass (PDM) models admitting deformed kinetic terms and modified commutators (Costa et al., 2021, Takou et al., 10 Aug 2025), noncommutative plane realizations via deformation quantization (Jim et al., 3 May 2024), and sine-square or $f(x)$-deformation through explicit spatial weights (Okunishi et al., 2015, Quesne, 2020).
• Supersymmetric Index Deformations: Formulations related to weak or higher-order supersymmetry, often employing matrix representations with nonstandard parity assignments to capture enriched index structures (Spiridonov, 16 Apr 2024, Spiridonov, 17 Jul 2024).

2. Deformation-Induced Regularization and Isospectrality Restoration

A central occurrence in DSQM is the breakdown of isospectrality between SUSY partner Hamiltonians in the presence of singularities. For standard realizations,

$$H^{(U)} = BA, \quad H^{(L)} = AB, \quad\text{with}\quad A = -\frac{d}{dx} + \mathcal{W}(x),\quad B = +\frac{d}{dx} + \mathcal{W}(x).$$

When $\mathcal{W}(x)$ or $V(x)$ is singular, spectral correspondence may be lost. The quantum toboggan model regularizes such singularities by deforming the coordinate integration contour to a multi-sheeted, winding path in the complex plane, thereby "bypassing" branch points and pathological short-distance behavior (Znojil, 2011). This process is often accompanied by a planarization—the mapping of the multisheet Riemann surface onto a single-valued complex domain via explicit changes of variables (e.g., $x \sim y^\alpha$).

Consequently, the resulting planarized systems can frequently be cast in the form of a weighted Sturm-Schrödinger equation,

$H^{(2)}\psi(y) = E\, W(y)\psi(y),$

with a nonconstant weight $W(y)$, enabling the use of standard spectral theory and restoration of isospectral correspondence—even for initially highly singular partner potentials.

3. Extended and Deformed Superalgebras

Deformations can transform the underlying supersymmetry algebra:

Deformation Type	Algebraic Structure	Spectral/Multiplet Consequence
$q$-Deformation (TD-type, scaling operators)	$B = (1/\sqrt{2})[T_q p - iW(x)]$	Non-Gaussian ground states; no standard level pairing (Gavrilik et al., 2013)
Higher-derivative (e.g. parasupersymmetry)	$(Q^\pm)^n = 0$ for $n > 2$; polynomials in $H$	Polynomial superalgebra; altered BPS structure (Spiridonov, 16 Apr 2024, Spiridonov, 17 Jul 2024)
Deformed/centrally extended supergroups	$SU(2|1)$, $SU(2|2)$, $OSp(4|2)$	Degeneracy patterns reflect representation content, hidden supersymmetries; energy levels classified by e.g. $q$ (Ivanov et al., 2013, Fedoruk et al., 2017, Sidorov, 2019)
Deformed shape invariance (via $f(x)$, $\hbar$)	Infinite systems of PDEs replacing difference relations	Nontrivial rational extensions and exact solvability in new families of potentials (Quesne, 2020)

In particular, the emergence of hidden (often larger) supersymmetries, e.g. from $SU(2|1)$ to $SU(2|2)$, organizes energy eigenstates into richer multiplets, explaining the presence of degeneracies beyond those predicted by the manifest algebra (Ivanov et al., 2013, Sidorov, 2019). Exceptional cases exist, such as "long multiplets" (Ivanov et al., 2015), where nontrivial couplings defined by deformation parameters prevent the multiplet from decomposing into short, decoupled chiral units.

4. Discretizations, Non-Hermiticity, and Noncommutative Models

DSQM extends to discrete and non-Hermitian settings:

• Lattice (Discretized) SUSY QM: Discretization of supersymmetric quantum mechanics, typically with $N=2$, leads to natural separation of the partition function into bosonic and fermionic sectors, accompanied by exact transfer matrix representations. The careful handling of lattice artifacts is essential for accurate restoration of continuum supersymmetry, particularly in the presence of spontaneous breaking (goldstino modes) and sign problems in the Witten index (Baumgartner et al., 2012).
• Non-Hermitian and $\mathcal{PT}$-Symmetric Deformations: Many deformations lead to non-Hermitian Hamiltonians in a "false" Hilbert space, which, upon suitable metric redefinition, can be rendered self-adjoint in a "physical" space. Deformations via complexification (including quantum toboggans) are closely connected with advances in $\mathcal{PT}$-symmetric quantum theory (Znojil, 2011).
• Deformation Quantization and Noncommutative Geometry: SUSY QM on a noncommutative plane is constructed using deformation quantization, with observables as elements of a noncommutative involutive algebra endowed with a star-product $*^r$ (Jim et al., 3 May 2024). The resulting spectra and Witten index remain independent of the noncommutativity parameter and deformation gauge, demonstrating that supersymmetry (including nontrivial ground states) survives such deformations.

5. Exact Solvability, Integrability, and Spectral Analysis

Exact analyses in DSQM frequently leverage:

• Deformed Shape Invariance: A generalization of the shape invariance technique replaces the difference-differential SI condition by infinite sets of PDEs, allowing construction and classification of exactly solvable deformed potentials and their spectrum via parameter shifts and deformations (Quesne, 2020).
• Sine-Square and Geometric Deformations: The sine-square deformation (SSD) modifies spatial envelopes ($f(x) = \sin^2(\pi x/L)$), yet—via SUSY QM and shape invariance—leads to ground states identical to the uniform (periodic) case, with excited states altered by the real-space deformation. Lattice-continuum correspondence requires geometric (metric) correction factors in wavefunctions (Okunishi et al., 2015).
• Exact WKB, Resurgence, and TBA Equations: For deformed systems with polynomial superpotentials and explicit $\hbar$-corrections, the spectral problem can be analyzed by exact WKB methods and interpreted via $\mathbb{Z}_4$-extended thermodynamic Bethe ansatz (TBA) systems. These approaches incorporate wall-crossing phenomena in moduli space, preserving the underlying integrability and $\mathbb{Z}_4$ structures through nontrivial analytic continuations (Ito et al., 8 Jan 2024, Ito et al., 8 Oct 2025). For instance, the cubic superpotential yields a TBA splitting into two $D_3$-type systems upon analytic continuation across critical walls.

Model/Technique	Deformation Parameter	Analytical Tool	Key Outcome
Quantum Toboggan (QTM)	Multisheeted contour	Planarization, rectification	Regularized singularities, Sturm-Schrödinger form
$q$-Deformed SQM (TD-type)	$q$	q-scaling, bibasic hypergeometric	Non-Gaussian gs, new degeneracy patterns
SSD / $f(x)$-deformation	Spatial envelope $f(x)$	SUSY QM, shape invariance	Ground state coincidence, metric correction
Deformed shape invariance	$\hbar$ or $f(x)$	PDE expansion	New exactly solvable families
TBA/WKB integrability	$\hbar$-corrections, $m$	ODE/IM correspondence	Nonperturbative spectrum, wall-crossing behavior

6. Multiplet Structure, Indices, and Hidden Symmetries

Deformed models give rise to a hierarchy of multiplet structures:

• Parasupersymmetric and Weak Supersymmetry Models: Multiplets larger than the standard two-component (or two-state) structure arise (e.g., 3×3 or 4×4 matrix algebras). These can be interpreted as higher-derivative, parafermionic, or centrally-extended models (Spiridonov, 16 Apr 2024, Spiridonov, 17 Jul 2024). Superconformal indices, computed in matrix models with extra operators and nontrivial central extensions, coincide with generalized Witten indices, accounting for vacuum sector richness and BPS counting.
• Hidden or Enhanced Supersymmetries: In models with supergroup symmetry (e.g., $SU(2|1)$, $SU(2|2)$, $OSp(4|2)$), the physical state space at each energy level is often spanned by irreducible representations of a "hidden" or larger supersymmetry, rather than the manifest algebraic symmetry (Ivanov et al., 2013, Sidorov, 2019, Fedoruk et al., 2017). This underlies nontrivial degeneracies and connects to superconformal index computations in field theory (Spiridonov, 16 Apr 2024).
• Witten Index and BPS Counting: In all deformations, careful analysis of the Witten index (or its variants like the superconformal index) is essential to determine spontaneous supersymmetry breaking, vacuum sector structure, and the robustness of SUSY under deformations (Baumgartner et al., 2012, Spiridonov, 16 Apr 2024, Jim et al., 3 May 2024, Spiridonov, 17 Jul 2024). In systems with topological (graph-theoretic) deformations, the index is linked to invariants such as the Euler characteristic, with spontaneous SUSY breaking prevented in discrete models without "null" (empty) graphs (Herz et al., 10 Jul 2025).

7. Physical Applications and Theoretical Impact

Deformed supersymmetric quantum mechanics provides broad theoretical and applied benefits:

• Singularity Regularization and Non-Hermitian Quantum Theory: QTM and complex deformations allow precise treatments of singular potentials that conventionally break isospectrality and SUSY. The results connect to the field of non-Hermitian (especially $\mathcal{PT}$-symmetric) quantum mechanics, and have implications for quantum optics and open systems (Znojil, 2011).
• Exactly Solvable Models and Special Functions: Deformed shape invariance methods and non-standard algebraic constructions generate new classes of solvable potentials, often associated with exceptional or deformed orthogonal polynomials, and underpin the theory of elliptic hypergeometric functions central to supersymmetric index computations (Quesne, 2020, Spiridonov, 16 Apr 2024).
• Integrable Systems and Wall-Crossing: DSQM with polynomial superpotentials and $\hbar$-deformations interpolates between ODE and integrable model (IM) phenomena, unifying concepts from spectral theory, resurgence, TBA systems, and wall-crossing in moduli space (Ito et al., 8 Jan 2024, Ito et al., 8 Oct 2025).
• Quantum Information and Entanglement: Energy-dependent or "unconventional" supersymmetry in spin systems connects exactly solvable models (Jaynes–Cummings, Rabi) and matrix product state representations, providing tools for understanding entanglement structure in more complex quantum devices (Naseri et al., 2020).
• Noncommutative and Discrete Quantum Geometry: Gauge-invariant deformation quantization and graph-theoretic SQM extend the reach of SUSY QM to noncommutative spaces and discrete systems, preserving key spectral and topological features in these contexts (Jim et al., 3 May 2024, Herz et al., 10 Jul 2025).

References to Key Mathematical Structures

• Factorization in SUSY with deformation:

$$A = -\mathcal{T} \frac{d}{dx} + \mathcal{T}\mathcal{W}(x), \quad B = \frac{d}{dx}\mathcal{T}^{-1} + \mathcal{W}(x)\mathcal{T}^{-1}$$

• Planarized (rectified) Sturm–Schrödinger form:

$H^{(2)} \psi(y) = E W(y) \psi(y)$

• $q$-deformed operator:

$$B = \frac{1}{\sqrt{2}}(T_q\,\hat{p} - i W(x)),\quad T_q f(x) = f(qx)$$

• SSD and shape invariance:

$$H_\beta = -\frac{d^2}{dx^2} + \frac{(\pi/L)^2 \beta(\beta-1)}{\sin^2(\pi x/L)},\quad A_\beta = \frac{d}{dx} - \beta \frac{\pi}{L} \cot\left(\frac{\pi x}{L}\right)$$

• Deformed shape invariance condition:

$$W^2(x, a) + h f(x) W_x(x, a) + g(a) = W^2(x, a+h) - h f(x) W_x(x, a+h) + g(a+h)$$

• Generalized commutator for PDM models:

$[x, \Pi_\gamma] = i\hbar(1 + \gamma x)$

• Wall-crossing TBA equation kernel:

$$K_\pm(\theta) = \frac{1}{4\pi}\left[\frac{1}{\cosh\theta} \pm i \frac{\sinh\theta}{\cosh\theta}\right]$$

Summary

Deformed supersymmetric quantum mechanics encompasses a family of models and analytic techniques in which algebraic, topological, or geometric deformities are systematically implemented to regularize singularities, enrich spectral structures, or adapt to complex geometries. These constructions generalize standard SUSY QM, enable new exactly solvable models, unify spectral problems with integrable systems via the ODE/IM correspondence and exact WKB analysis, and preserve or explain the persistence of unbroken SUSY and robust vacuum counting even in non-Hermitian, noncommutative, or discrete settings. They have opened crucial connections between quantum mechanics, special function theory, quantum field theory indices, and integrable systems, serving as an active area bridging mathematical physics, algebraic geometry, and quantum information.