Comment on “Self-Isospectral Periodic Potentials and Supersymmetric Quantum Mechanics”

We show that the formalism of supersymmetric quantum mechanics applied to the solvable elliptic function potentials V (x) = mj(j + 1)sn(x,m) produces new exactly solvable onedimensional periodic potentials. In a recent paper, Dunne and Feinberg [1] have systematically discussed various aspects of supersymmetric quantum mechanics (SUSYQM) as applied to periodic potentials. In particular, they defined and developed the concept of self-isospectral periodic potentials at length. Basically, a one dimensional potential V−(x) of period 2K is said to be self-isospectral if its supersymmetric partner potential V+(x) is just the original potential upto a discrete transformation a translation by any constant amount, a reflection, or both. An example is translation by half a period, that is V+(x) = V−(x−K). In this sense, a self-isospectral potential is somewhat trivial, since application of the SUSYQM formalism [2] to it yields nothing new. The main example considered in ref. [1] is the class of elliptic function potentials V (x) = mj(j + 1)sn(x,m) , j = 1, 2, 3, . . . (1) Here sn(x,m) is a Jacobi elliptic function of real elliptic modulus parameter m (0 ≤ m ≤ 1). From now on, for simplicity, the argument m is suppressed. The Schrödinger equation of the