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We discuss in some detail the self-similar potentials of Shabat and Spiridonov which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape invariant potentials of which the Shabat-Spiridonov ones are a special case.These new potentials can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions and transmission coefficients for these potentials are obtained. We show that these potentials can also be obtained numerically. Included as an intriguing case is a shape invariant double well potential whose supersymmetric partner potential is only a single well. Our class of exactly solvable Hamiltonians is further enlarged by examining two new directions: (i) changes of parameters which are different from the previously studied cases of translation and scaling; (ii) extending the usual concept of shape invariance in one step to a multi-step situation. These extensions can be viewed as q-deformations of the harmonic oscillator or multi-soliton solutions corresponding to the Rosen-Morse potential.(

Starting from a potential with a continuum of energy eigenstates, we show how the methods of supersymmetric quantum mechanics can be used to generate families of potentials with bound states in the continuum [BICs]. We also find the corresponding wave functions. Our method preserves the spectrum of the original potential except it adds these discrete BICs at selected energies. Specifically, we compute and graph potentials which have bound states in the continuum starting from a null potential representing a free particle and from both the attractive and the repulsive Coulomb potentials.

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We give a relativistic description of composite particles which behave as nonrelativistic clusters in their rest frames. We find that the relativistic form factor can be obtained from the nonrelativistic one by the substitution S(t) =oc("n)'2So(ltl /a), where a=l-t/(4Ma2), and where n is the number of subparticles in the clusier. An excellerit one-parameter fit to ihe magnetic iorm factor of the proton is found.