Deformed Shape Invariance in Quantum Systems

• Deformed shape invariance is a symmetry condition where the functional form of SUSY partner potentials is preserved through a deformation-dependent parameter shift.
• It is applied to modify the Schrödinger equation in systems with position-dependent mass and curved-space metrics, enabling exact and extended solutions.
• The concept extends to geometrical and statistical shape analysis, yielding canonical forms and alignment-free models that remain stable under nonrigid transformations.

Deformed shape invariance denotes a family of symmetry constructions in which either a quantum system preserves the functional form of its SUSY partner under a deformation-dependent parameter shift, or a geometric or statistical representation is designed to remain stable under a specified class of nonrigid transformations. In supersymmetric quantum mechanics the term is used explicitly in deformed supersymmetric frameworks with a positive deforming function $f$, position-dependent mass, or constant-curvature geometry (Quesne, 2016, Quesne, 2020). In geometry, computer vision, and shape analysis, closely related work studies invariants or canonical forms under elastic, affine, conformal, diffeomorphic, or learned deformation models (Arabadjis et al., 2012, Raviv et al., 2010, Hossain et al., 2024).

1. Deformed shape invariance in deformed supersymmetric quantum mechanics

In the curved-space and position-dependent-mass setting, the radial Schrödinger problem is rewritten as a deformed Schrödinger equation,

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$

with the equivalent position-dependent mass

$m(r)=\frac1{f^2(r)}.$

The corresponding first-order DSUSY operators are

$$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$

and the partner potentials are

$$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$

Deformed shape invariance is then the statement that

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$

or equivalently

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$

This is the curved-space analogue of ordinary shape invariance, but with the derivative and factorization scheme modified by the deformation function $f$ (Quesne, 2016).

A parallel formulation is given in deformed SUSY quantum mechanics with translational parameter shift $a_{i+1}=a_i+\hbar$. There the deformed first-order operators are

$$A^\pm(a)= \mp \hbar \sqrt{f(x)}\frac{d}{dx}\sqrt{f(x)} + W(x,a),$$

with

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$0

and the deformed shape-invariance condition becomes

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$1

In this formulation the deformation admits the interpretation of a position-dependent mass system with $$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$2, or equivalently a curved-space problem with diagonal metric $$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$3 (Quesne, 2020).

2. Exact, conditional, and enlarged forms

For the nonlinear harmonic oscillator in constant-curvature space, the reduced deformed radial potential is

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$4

with superpotential

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$5

Its partner satisfies

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$6

so the parameter shift is

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$7

For the curved Kepler–Coulomb problem,

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$8

and the partner relation is

$$\left(-\sqrt{f(r)}\,\frac{d}{dr}\,f(r)\,\frac{d}{dr}\sqrt{f(r)}+V(r)-E\right)\psi(r)=0, \qquad f(r)=\sqrt{1+\lambda r^2},$$9

These are the basic exact DSI examples in the constant-curvature setting (Quesne, 2016).

The PDE methodology for translational DSI replaces the difference-differential condition by local equations for the superpotential. For $m(r)=\frac1{f^2(r)}.$0-independent $m(r)=\frac1{f^2(r)}.$1, the hierarchy reduces to

$m(r)=\frac1{f^2(r)}.$2

together with

$m(r)=\frac1{f^2(r)}.$3

For explicitly $m(r)=\frac1{f^2(r)}.$4-dependent superpotentials,

$m(r)=\frac1{f^2(r)}.$5

the deformed shape-invariance condition generates an infinite set of recursive PDEs. This extended method is illustrated on deformed Pöschl–Teller, radial harmonic oscillator, Scarf I, Coulomb, Morse, Eckart, Rosen–Morse I, shifted harmonic oscillator, deformed radial harmonic oscillator, and deformed Coulomb systems, and also on a rationally extended deformed radial oscillator (Quesne, 2020).

A more restrictive phenomenon is conditionally deformed shape invariant behavior. For two families of extensions of the oscillator in a $m(r)=\frac1{f^2(r)}.$6-dimensional constant-curvature space, the first two members are deformed-shape-invariant only if constraint conditions relating the potential parameters are satisfied. Because the constraint conditions change in the second SUSY step, compatibility conditions between the two sets are imposed to build potentials with known ground and first excited states. The general construction is then recast in terms of a generating function $m(r)=\frac1{f^2(r)}.$7 and an accompanying $m(r)=\frac1{f^2(r)}.$8, from which the first two superpotentials, the first two partner potentials, and the first two eigenstates are obtained (Quesne, 2017).

The rationally extended curved-space systems exhibit a further variant, enlarged deformed shape invariance. For isospectral type-I and type-II rational extensions of the hyperbolic Kepler–Coulomb problem, the partner relation becomes

$m(r)=\frac1{f^2(r)}.$9

so shape invariance acts not only on the physical parameter $$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$0 but also on the polynomial degree $$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$1 appearing in the rational denominator. This is the paper’s strongest generalization of DSI beyond the nonextended case (Quesne, 2016).

3. Algebraic, phase-space, and higher-dimensional reformulations

Shape invariance can be recast as a nonlinear potential algebra. Introducing an auxiliary periodic variable $$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$2, one promotes the parameter to

$$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$3

and defines

$$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$4

so that

$$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$5

For the flat, spherical, and hyperbolic Kepler problems and for the Rosen–Morse potential, the nontrivial commutator $$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$6 becomes a nonlinear, non-polynomial function of $$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$7. The undeformed limits reduce to $$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$8, $$\hat A^\pm(\boldsymbol\mu)=\mp \sqrt{f(r)}\,\frac{d}{dr}\,\sqrt{f(r)}+W(r;\boldsymbol\mu),$$9, or $$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$0, but for $$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$1 the algebra closes through rational structure functions such as

$$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$2

This does not introduce deformed shape invariance in the curved-space DSUSY sense, but it reveals a nonlinear algebraic content of ordinary shape invariance (1711.02422).

A different reformulation is the phase-space version based on deformation quantization. Operators are replaced by Weyl symbols, and operator multiplication becomes the Groenewold–Moyal star product

$$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$3

The stationary-state equation becomes the star-genvalue equation

$$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$4

with $$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$5 a Wigner function. In this setting ordinary shape invariance induces the phase-space relation

$$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$6

and yields the recursion

$$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$7

The deformation here is the $$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$8-deformation of the classical phase-space algebra, not the DSUSY deformation induced by $$V_0(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)-f(r)W'(r;\boldsymbol\mu)+\epsilon_0, \qquad V_1(r;\boldsymbol\mu)=W^2(r;\boldsymbol\mu)+f(r)W'(r;\boldsymbol\mu)+\epsilon_0.$$9 (Rasinariu, 2012).

Higher-dimensional generalization is also possible without using the term “deformed” in the narrow sense. In polynomial two-dimensional SUSY quantum mechanics with second-order supercharges,

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$0

the construction uses additive shape invariance

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$1

and produces non-separable two-dimensional generalized Morse and generalized Pöschl–Teller systems with fourth-order symmetry operators. This is a higher-dimensional generalization of additive shape invariance rather than a deformed-shape-invariance construction in the constant-curvature DSUSY sense (Cannata et al., 2011).

A useful boundary case is the reported failure of ordinary shape invariance. A “new superpotential” has been proposed for which “neither the supersymmetric energy conditions nor the associated shape invariance condition remain valid,” while a new energy condition

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$2

emerges between the partner Hamiltonians $$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$3. The abstract further states that when the superpotential is associated with discontinuity or distortion, “SUSY energy conditions and the shape invariance condition will no longer hold good” (Dong et al., 2022). This suggests that deformation may also act as a mechanism for the breakdown of shape invariance, not only for its generalization.

4. Deformation-invariant geometry and canonical shape recovery

Outside SUSY quantum mechanics, closely related constructions use controlled deformations to define invariants or canonical representations. In 2D shape characterization via boundary distortion, the descriptor begins with

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$4

and is enriched by area-preserving anisotropic distortions

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$5

where $$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$6 expands in one direction and contracts in the orthogonal direction with determinant $$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$7. The resulting representation is invariant to translation, rotation, reflection, and scaling after quotienting, but it is not invariant to the anisotropic distortions themselves; those distortions are the probe family used to characterize shape response (Descombes et al., 2013).

Affine differential geometry supplies a second route. Replacing the Euclidean metric by an equi-affine-invariant tensor

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$8

allows standard geodesic and spectral machinery to operate in an equi-affine-invariant metric. One paper uses this metric to compute affine-invariant geodesic distances by fast marching, while a companion paper uses the same metric to define an affine-invariant Laplace–Beltrami operator, heat kernel, diffusion distance, and commute-time distance. In both cases the exact invariance class is global equi-affine transformations, not arbitrary nonrigid deformation (Raviv et al., 2010, Raviv et al., 2010).

Conformal invariance provides a third class of explicit formulas. For 2D Möbius transformations the differential expression

$$\hat A^-(\boldsymbol\mu_i)\hat A^+(\boldsymbol\mu_i) = \hat A^+(\boldsymbol\mu_{i+1})\hat A^-(\boldsymbol\mu_{i+1})+\epsilon_{i+1},$$9

is stated to be an absolute invariant. For 3D conformal or Möbius transformations the corresponding invariant is

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$0

with

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$1

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$2

Integral invariants are then built from these relative and absolute invariants (Zhang et al., 2018).

A mechanically grounded canonicalization appears in the methodology for 2D elastic bodies. Under the assumptions of isotropic, homogeneous, continuous material, static balance, linear stress–strain relation, and cross sections that remain straight and perpendicular to a neutral line, the authors derive that the neutral line is the image of the undeformed reference line, that no stress is exerted along it, that it has the same length as the undeformed reference line, that it passes through the midpoint of each cross section, and that cross sections remain perpendicular to it. These invariants support two inversion procedures: one based on neutral-line and cross-section recovery, and another based on solving deformation PDEs by equivalent image operations. In the parasite application, the resulting unwrapped contours enabled automatic classification into 6 families “with an accuracy rate of 97.6 %” (Arabadjis et al., 2012).

5. Alignment-free statistical models and deformable registration frameworks

A different line of work treats invariance as a property of the representation space rather than of explicit differential formulas. In alignment-free statistical shape modeling based on discrete fundamental forms, a surface is represented by local metric distortion

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$3

and transition rotations

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$4

yielding a product-Lie-group descriptor

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$5

The representation is invariant under global rigid motions because translations are removed by passing to deformation gradients and rotations cancel in the relative quantities $$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$6. It is not invariant to nonrigid deformation itself; rather, it represents deformation through discrete first and second fundamental forms in a way that remains alignment-free (Ambellan et al., 2021).

In computational anatomy, shape analysis with diffeomorphisms is related to symmetry through local involutions and transvections. With LDDMM registration, one writes a target shape as

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$7

and defines a residual-corrected symmetry

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$8

The midpoint is similarly corrected,

$$W^2(r;\boldsymbol\mu_i)+f(r)W'(r;\boldsymbol\mu_i) = W^2(r;\boldsymbol\mu_{i+1})-f(r)W'(r;\boldsymbol\mu_{i+1})+\epsilon_{i+1}.$$9

The centrality, involutivity, and transvectivity defects then measure how far the deformation space is from exact symmetric-space behavior. This does not eliminate deformation; it quantifies the degree to which deformation-induced operations behave like invariant geometric symmetries (Guigui et al., 2019).

Deformable Generalized Procrustes Analysis extends canonical alignment to nonrigid transformations. For Linear Basis Warps,

$f$0

the latent reference shape $f$1 is constrained by

$f$2

where $f$3 fixes the eigenvalues of the shape covariance. Under the free-translation property, the reduced GPA problem admits a closed-form globally optimal solution by eigenvalue decomposition. This makes the canonical shape invariant to translation and rigid gauge freedoms while allowing nonrigid TPS- or affine-type deformations to be estimated explicitly (Bai et al., 2022).

6. Learning-based usage, broader analogues, and limits of the concept

In learning-based image analysis, “invariant shape representation learning” has a more specific meaning. ISRL represents each sample by a deformation-based latent variable, the initial velocity field $f$4 of a diffeomorphic registration model governed by EPDiff, and then enforces invariant risk minimization across environments. The classifier is called invariant when “there exists a model $f$5 simultaneously optimal for all environments.” Here invariance is not invariance to deformation itself; deformation is the medium of representation, while invariance is sought with respect to environment-specific, spurious correlations (Hossain et al., 2024).

A broader but adjacent usage appears in nuclear-structure studies. In the proxy-SU(3) model, once the SU(3) irrep $f$6 is fixed, the shape variables are given analytically by

$f$7

$f$8

with $f$9 controlling the deformation amplitude. The paper explicitly notes that it “does not use that phrase,” but it is relevant because gross shape properties are produced by symmetry invariants rather than by local fitting (Bonatsos et al., 2017). By contrast, in deformed shell-model calculations of double beta decay, the observables are not shape invariant: the paper states that the $a_{i+1}=a_i+\hbar$0 and $a_{i+1}=a_i+\hbar$1 nuclear transition matrix elements “will reduce considerably as we go from spherical shapes to deformed shapes” (Sahu et al., 2015). A plausible implication is that “shape invariance” and “shape determined by symmetry” should be kept distinct.

Several recurrent misconceptions are therefore ruled out by the literature. Deformed shape invariance is not a universal synonym for arbitrary nonrigid invariance; in DSUSY it refers to a specific deformation of the factorization and Riccati structure (Quesne, 2016, Quesne, 2020). In phase-space work, the deformation is the Moyal $a_{i+1}=a_i+\hbar$2-product, not the curved-space $a_{i+1}=a_i+\hbar$3-deformation (Rasinariu, 2012). In machine learning, invariance may mean invariance across environments while deformation remains explicitly represented (Hossain et al., 2024). In geometry-processing and elastic-body reconstruction, the decisive object is often a canonical undeformed representation rather than a quotient that removes all deformation information (Arabadjis et al., 2012, Descombes et al., 2013). Across these settings, the unifying theme is not a single formal definition but a shared program: identify the part of shape change that should be treated as nuisance, encode the part that should remain explicit, and formalize the resulting balance through deformed partner relations, invariant metrics, canonical coordinates, or environment-stable latent variables.