Conditions for nondegeneracy in supersymmetric quantum mechanics

Tom D. Imbo; Uday P. Sukhatme

doi:10.1103/PhysRevD.33.3147

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Conditions for nondegeneracy in supersymmetric quantum mechanics

Tom D. Imbo and Uday P. Sukhatme

Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60680

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Phys. Rev. D 33, 3147 – Published 15 May, 1986

DOI: https://doi.org/10.1103/PhysRevD.33.3147

Abstract

It is shown that the positive ‘‘bosonic’’ and ‘‘fermionic’’ bound-state spectra in spherically symmetric supersymmetric (SUSY) quantum mechanics are degenerate if and only if the superpotential W(r) satisfies δ≡ $\lim_{r \to 0}$ r‖W(r)‖≥0.5. Also, if δ<0.5, then SUSY is broken.

References (10)

P. Roy and R. Roychoudhury, Phys. Rev. D 32, 1597 (1985).
E. Witten, Nucl. Phys. B188, 513 (1981).
For a review, see, for example, F. Cooper and D. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983).
We require a physical solution of the reduced radial Schrödinger equation to vanish as r -> 0 at least as fast as $r^{0.5}$ (in addition to vanishing at large r). This guarantees that the full Hamiltonian is self-adjoint with respect to the space of its physical solutions (or, equivalently, that such solutions have finite energy). This is in contrast with the case of a one-dimensional Hamiltonian with a potential V ( x ) where imposition of the above finite-energy condition again forces all physical solutions of the Schrödinger equation to vanish as x -> 0 at least as fast as $x^{0.5}$ , except in the case when $\lim_{x - > 0}$ x V ( x ) = 0, where physical solutions which go to a non\%zero constant (with zero slope) at x = 0 are also allowed.
Such states are not even physical solutions of the one-dimensional potentials $\tilde{V}$ ${}_{+ -}$ ( x ) (see Ref. 4).
In this case, there is no degeneracy even between the physical spectra of $\tilde{V}$ ${}_{+ -}$ ( x ) ; i.e., the degeneracy theorem does not hold even in the one-dimensional SUSY system.
The case of δ = 1 is interesting. For this case [which does not correspond to any of situations (1)–(3)] the one-dimensional SUSY system $\tilde{V}$ ${}_{+ -}$ ( x ) in general admits negative-energy states, and normalizable solutions of one potential may be ``paired'' by the degeneracy theorem with non-normalizable solutions of the other. This happens because in this situation the (one-dimensional) operators $Q^{+ -}$ are not adjoints of each other in the manner which is assumed in the proof of the non-negativity of the energy and the pairing of normalizable states. However, these problems occur in such a way as to yield a sensible SUSY system when one considers the half-potentials $V_{+ -}$ ( r ).
The case δ = inf corresponds to so-called repulsive ``singular'' potentials $V_{+ -}$ ( r ) [see W. Frank, D. Land and R. Spector, Rev. Mod. Phys. 43, 36 (1971)]. It is interesting that one can never generate attractive singular potentials in SUSY quantum mechanics. Such potentials lead to problems of physical interpretation (see Frank, Land, and Spector).
It is also common to use a superpotential constructed from the physical ground-state wave function $ψ_{0}$ of a given Hamiltonian $H_{+}$ , W ( r ) = ψ ${}_{0 ~}$ ${}^{sprime}$ / $ψ_{0}$ . This is often called the ``method of factor- ization'' {see, e.g., A. Adrianov, N. Borisov and M. Ioffe, Pis'ma Zh. Eksp. Teor. Fiz. 39, 78 (1984) [JETP Lett. 39, 93 (1984)], and references therein} or the ``ground-state wave-function representation'' [E. Gozzi, Phys. Lett. 129B, 432 (1983)]. For such systems, one is assured that the degeneracy theorem holds for $V_{+ -}$ ( r ) (i.e., δ >eq 0.5 ), and that SUSY is unbroken.
Connections between supersymmetry breaking and physically acceptable states have also been discussed by A. Jevicki and J. P. Rodrigues, Phys. Lett. 146B, 55 (1984) and J. Fuchs, J. Math. Phys. 27, 349 (1986).

[c1] P. Roy and R. Roychoudhury, Phys. Rev. D 32, 1597 (1985).

[c2] E. Witten, Nucl. Phys. B188, 513 (1981).

[c3] For a review, see, for example, F. Cooper and D. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983).

[c4] We require a physical solution of the reduced radial Schrödinger equation to vanish as r -> 0 at least as fast as $r^{0.5}$ (in addition to vanishing at large r). This guarantees that the full Hamiltonian is self-adjoint with respect to the space of its physical solutions (or, equivalently, that such solutions have finite energy). This is in contrast with the case of a one-dimensional Hamiltonian with a potential V ( x ) where imposition of the above finite-energy condition again forces all physical solutions of the Schrödinger equation to vanish as x -> 0 at least as fast as $x^{0.5}$ , except in the case when $\lim_{x - > 0}$ x V ( x ) = 0, where physical solutions which go to a non\%zero constant (with zero slope) at x = 0 are also allowed.

[c5] Such states are not even physical solutions of the one-dimensional potentials $\tilde{V}$ ${}_{+ -}$ ( x ) (see Ref. 4).

[c6] In this case, there is no degeneracy even between the physical spectra of $\tilde{V}$ ${}_{+ -}$ ( x ) ; i.e., the degeneracy theorem does not hold even in the one-dimensional SUSY system.

[c7] The case of δ = 1 is interesting. For this case [which does not correspond to any of situations (1)–(3)] the one-dimensional SUSY system $\tilde{V}$ ${}_{+ -}$ ( x ) in general admits negative-energy states, and normalizable solutions of one potential may be ``paired'' by the degeneracy theorem with non-normalizable solutions of the other. This happens because in this situation the (one-dimensional) operators $Q^{+ -}$ are not adjoints of each other in the manner which is assumed in the proof of the non-negativity of the energy and the pairing of normalizable states. However, these problems occur in such a way as to yield a sensible SUSY system when one considers the half-potentials $V_{+ -}$ ( r ).

[c8] The case δ = inf corresponds to so-called repulsive ``singular'' potentials $V_{+ -}$ ( r ) [see W. Frank, D. Land and R. Spector, Rev. Mod. Phys. 43, 36 (1971)]. It is interesting that one can never generate attractive singular potentials in SUSY quantum mechanics. Such potentials lead to problems of physical interpretation (see Frank, Land, and Spector).

[c9] It is also common to use a superpotential constructed from the physical ground-state wave function $ψ_{0}$ of a given Hamiltonian $H_{+}$ , W ( r ) = ψ ${}_{0 ~}$ ${}^{sprime}$ / $ψ_{0}$ . This is often called the ``method of factor- ization'' {see, e.g., A. Adrianov, N. Borisov and M. Ioffe, Pis'ma Zh. Eksp. Teor. Fiz. 39, 78 (1984) [JETP Lett. 39, 93 (1984)], and references therein} or the ``ground-state wave-function representation'' [E. Gozzi, Phys. Lett. 129B, 432 (1983)]. For such systems, one is assured that the degeneracy theorem holds for $V_{+ -}$ ( r ) (i.e., δ >eq 0.5 ), and that SUSY is unbroken.

[c10] Connections between supersymmetry breaking and physically acceptable states have also been discussed by A. Jevicki and J. P. Rodrigues, Phys. Lett. 146B, 55 (1984) and J. Fuchs, J. Math. Phys. 27, 349 (1986).

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Conditions for nondegeneracy in supersymmetric quantum mechanics

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