By Mickelsson J.
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The unnormalized ground state wave function is $o (1) 22 ( x ; a l )= exp[--(rl 2 x4 + r2)+ $d2r: + 2d3r1r2+ d&) + o ( x S ) ]. 8). 10) and the fact that for this SIP a2 = qal. 67) where + + + W(oo,Uj) = @'- E y . 68) Now, as n + oo,an+l = qnal + O(0 < q < 1) and, since we have taken g o ( x ) = 0 , one gets W(x;a,+l) + 0. 69) The above discussion keeping only r1 ,r;! # 0 can be readily generalized to an arbitrary number of nonzero r j . 70) Strategies for Categorizing Shape Invariant Potentials 51 All these potentials are also symmetric and reflectionless with TI as given by eq.
However for the case n even and g arbitrary, there is no candidate matrix ground state wave function that is normalizable. In this case the potentials Vl and Vz have degenerate positive ground state energies and neither Q nor Qt annihilate the matrix ground state wave function as given by eq. 42). Thus we have the immediate result that if the ground state energy of the matrix Hamiltonian is non-zero then SUSY is broken. For the case of broken SUSY the operators A and At no longer change the number of nodes and there is a 1-1pairing of all the eigenstates of HIand H 2 .
9; - g; + . . + g; = -(g; - 9; g; - g; n - -Vjng,m f C j - Ccm . , + (ngcm12 = nccm > + wjngc,, = cj 9; + gongem = GI. 29) V j E (2, . 32) V j E (2, . * 1 n}. 33) Vib) i=2 gj(z) = gcm(z) + u j ( 2 ) t The interested reader is referred to the article by Cariiiena and Ramos for details. In general the answer for the superpotential can be given in terms of ratios of sums of sines and cosines or ratios of sums of sinh and cosh. It is suspected, however, that the solutions found for n 2 3 can be mapped into the solutions for n = 2 or n = 1 by a suitable change of parameters.
Advanced quantum mechanics by Mickelsson J.