Volume 142, number 1 PHYSICS LETTERS A 27 November 1989 IS SHAPE INVARIANCE ALSO NECESSARY FOR LOWEST ORDER SUPERSYMMETRIC WKB TO BE EXACT? Avinash KHARE Department ofPhysics, University of Illinois at Chicago, Chicago, USA and Institute ofPhysics, Sachivalaya Marg, Bhubaneswar 751005, India and Y.P. VARSHNI Ottawa-Carleton Institutefor Physics, University of Ottawa, Ottawa, Canada KIN 6N5 Received 8 June 1989; revised manuscript received 25 September 1989; accepted for publication 28 September 1989 Communicated by D.D. Hoim We study the bound state spectrum of two classes of exactly solvable non-shape-invariant potentials in the supersymmetric WKB (SWKB) approximation and show that it is not exact. These examples suggest that shape-invariance is not only sufficient but perhaps even necessary in order that the lowest order SWKB reproduces the exact bound state spectrum. Inspired by supersymmetry, about four years back recall the work of Gendenshtein [6] who proved that the SWKB formalism was suggested by Comtet et al. shape invariance is sufficient for a potential to be ex- [1]. It was immediately noticed by these and one of actly solvable and then conjectured that it may also the present authors [2] that the lowest order SWKB be necessary. However, Cooper et al. [7] proved that formalism reproduces the exact bound state spec- his conjecture is false by showing that the Natanzon trum for a number of exactly solvable potentials. The class of potentials [8] even though exactly solvable deeper reason for this was uncovered by Dutt et al. are not shape invariant. [3] when they showed that lowest order SWKB nec- The purpose of this Letter is to raise the interest- essarily reproduces the exact bound state spectrum ing but difficult question of whether shape invari- of any shape-invariant potential. Further, they [3] ance is (not only sufficient but) also necessary for as well as Adhikari et al. [4] have explicitly calcu- lowest order SWKB to be exact. At present one does lated 0(h 2) to 0(h6) corrections to the SWKB bound not know as to how to prove or disprove this con- state spectrum for all the known shape-invariant po- jecture. In the absence of that it might be worthwhile tentials and have shown that as expected they all to study the spectrum of some exactly solvable but vanish. Finally, using complex integration tech- non-shape-invariant potentials in lowest order SWKB nique, Raghunathan et al. [5] have shown that all approximation. The potential which immediately higher order SWKB corrections for the shape-invar- comes to mind is the Ginocchio potential [9] which iant Rosen—Morse potential are zero, is a special case of the Natanzon class of potentials. Thus shape invariance is clearly sufficient for low- Using the Ginocchio potential Cooper et al. [71 have est order SWKB to reproduce the exact bound state shown that even though shape invariance is suffi- spectrum (and for all higher order corrections to be cient it is not necessary for exact solvability of the zero). The obvious interesting question is whether Schrodinger equation. The other choice could be the shape invariance is also necessary for lowest order class of isospectral Hamiltonians corresponding to SWKB to be exact? In this context we would like to the various shape-invariant potentials. As a first step 0375-9601 f89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)