Breakup of SUSY Quantum Mechanics in the Limit-Circle Region of the Reflective Kratzer Oscillator (original) (raw)

The paper studies violation of conventional rules of SUSY quantum mechanics for the inverse-square-at-origin (IS@O) radial potential V(r) within the limit-circle (LC) range. A special attention is given to transformation properties of the Titchmarsh-Weyl m-function under Darboux deformations of the reflective Kratzer oscillator: centrifugal Kepler-Coulomb (KC) potential plus a Taylor series in r. Since our analysis is based on Fulton's representation of a regular-at-infinity (R@) solution [Math. Nachr. 281, 1418 (2008)] as a superposition of two Frobenius solutions at the origin, we refer to the appropriate expressions as the Titchmarsh-Weyl-Fulton (TWF) functions. Explicit transformation relations are derived for partner TWF functions associated with SUSY pairs of IS@O potentials. It is shown that these relations have a completely different form for Darboux transformations (DTs) keeping the potential within the LC range. As an illustration, we use regular nodeless Frobenius solutions to construct SUSY partners of the radial r-and c-Gauss-reference (GRef) potentials solvable via hypergeometric and confluent hypergeometric functions, respectively. We explicitly demonstrate existence of non-isospectral partners of both radial potentials in the LC region and obtain their discrete energy spectra using the derived closed-form expressions for the TWF functions. The general transformation relations for the TWF function have been verified taking advantage of form-invariance of the radial GRef potentials under double-step DTs with the so-called 'basic' seed solutions (SSs). Similarly we directly ratify that TWF functions for three shape-invariant reflective potentials on the half-line-hyperbolic Pöschl-Teller (h-PT), Eckart/Manning-Rosen (E/MR), and centrifugal KC potentialsdo retain their form under basic DTs.