[Paper Review] Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics

Sergei P. Maydanyuk|ArXiv.org|Oct 22, 2007

Quantum Mechanics and Non-Hermitian Physics62 references3 citations

TL;DR

This paper introduces a novel double complex supersymmetric quantum mechanics (SUSY) transformation method using non-degenerate complex energies to exactly deform real potentials while preserving their reality and spectral characteristics. By applying complex Darboux transformations with distinct energy parameters, the approach enables continuous, analytically tractable deformations of both discrete-spectrum (e.g., finite rectangular well) and continuous-spectrum (e.g., null potential) systems, yielding new isospectral, reflectionless, and asymmetric double-well potentials with full control over shape, depth, and symmetry—offering a simpler alternative to existing shape-invariant models.

ABSTRACT

In paper approach of double complex SUSY-transformations with not coincident complex energies of transformation is developed, allowing to deform given real potential $V_{1}$ with obtaining exact solutions. The explicit solutions of the deformation of shape of the potential, its wave function at arbitrary energy, not coincident with energies of transformation, wave functions at the energies of transformation are obtained, condition of keeping of continuity of the solutions and isospectral condition are determined. Using a rectangular well of finite width with infinitely high walls as the starting $V_{1}$ with discrete energy spectrum, by the proposed approach new types of deformation of this potential with deformation of the energy spectrum as a whole have been obtained. The new potential contains the rectangular well as own partial case (with simultaneous transformation of the shape of this new potential, energy spectrum, wave functions of all bound states, wave function at arbitrary energy into corresponding characteristics of the rectangular well at needed choice of parameters). Using null potential as the starting $V_{1}$ with continuous energy spectrum, new form of reflectionless real potential has been constructed. This potential generalizes well-known reflectionless potential of the type $V_{ m ref}(x) = A^{2}(1-2 { m sech}^{2}{Ax})$, allowing: to pull down tails of the potential $V_{ m ref}$ in the asymptotic regions up to zero (with keeping of nonzero depth); to pull down continuously the depth of the hole; to displace arbitrary along axis $x$ the hole with its passing through zero; to create and to increase the second hole, transforming $V_{ m ref}$ into double-well potential; to control continuously and simply the asymmetry of the shape of such reflectionless potential.

Motivation & Objective

• To develop a method for continuous, exact deformation of real quantum potentials while preserving their reality and spectral structure.
• To extend supersymmetric quantum mechanics beyond standard real-energy transformations by employing non-coincident complex energies of transformation.
• To construct new types of isospectral potentials, including deformed rectangular wells and generalized reflectionless potentials, with tunable asymmetry and depth.
• To provide explicit analytical solutions for wave functions at arbitrary energies and at transformation energies, ensuring continuity and spectral invariance.
• To demonstrate practical control over potential shape, including tail behavior, hole depth, and double-well formation, via simple parameter tuning.

Proposed method

• Utilizes double complex SUSY transformations with two distinct complex energies of transformation, denoted $\mathcal{E}_1$ and $\mathcal{E}_2$, to generate new potentials from a starting real potential $V_1$.
• Applies the factorization method and Riccati equation framework to derive superpotentials $W_i^{(k)}$ from solutions $\phi_i^{(k)}$ of the Schrödinger equation at transformation energies.
• Employs first-order Darboux transformations via operators $A_i = \frac{d}{dx} + W_i(x)$ and $A_i^+ = -\frac{d}{dx} + W_i(x)$ to generate isospectral partner potentials.
• Derives explicit closed-form expressions for the deformed potential $V_2(x)$, wave functions $\varphi_n^{(2)}$ at arbitrary energy, and $\varphi_m^{(2)}$ at transformation energies.
• Imposes continuity conditions on wave functions and ensures isospectrality by requiring the absence of divergences and discontinuities in the transformed solutions.
• Applies the method to two key starting potentials: (1) a finite rectangular well with infinite walls (discrete spectrum), and (2) the null potential (continuous spectrum), yielding new classes of potentials.

Experimental results

Research questions

• RQ1 Can double complex SUSY transformations with non-coincident complex energies generate new real potentials while preserving spectral isospectrality and solution continuity?
• RQ2 How can the shape of a finite rectangular well be continuously deformed via complex SUSY transformations, and what is the resulting behavior of its energy spectrum and wave functions?
• RQ3 Can the well-known reflectionless potential $V_{\rm ref}(x) = A^2(1 - 2\,{\rm sech}^2(Ax))$ be generalized using this method to allow tunable depth, asymmetry, and double-well formation?
• RQ4 What are the explicit analytical forms of wave functions at arbitrary energies and at transformation energies in the deformed potential?
• RQ5 To what extent can this method simplify the construction of reflectionless potentials compared to existing shape-invariant or self-similar models?

Key findings

• The method successfully generates new real potentials from a starting real potential $V_1$ without introducing divergences or discontinuities, ensuring continuity of wave functions.
• For the finite rectangular well, the approach produces a new potential that reduces to the original well in a limiting case, with all bound state wave functions and energy levels continuously deformable via parameter tuning.
• The null potential is transformed into a generalized reflectionless potential that allows continuous control over tail behavior (pulling down to zero), hole depth, position (including passage through zero), and the creation of a second hole, transforming into a symmetric double-well potential.
• By varying $\mathcal{E}_2$ from $-0.00001$ to $-19$, the method smoothly transforms a symmetric single-well potential into a symmetric double-well potential, demonstrating continuous asymmetry control.
• The resulting reflectionless potential is simpler in form than existing shape-invariant or self-similar reflectionless potentials, while retaining full reflectionless behavior and satisfying the construction rule from Ref. [69].
• The wave functions at arbitrary energies and at transformation energies are explicitly derived and shown to remain finite and continuous, confirming the validity of the isospectral transformation.