Abstract
We propose an extended version of supersymmetric quantum mechanics which can be useful if the Hamiltonian of the physical system under investigation is not Hermitian. The method is based on the use of two, in general different, superpotentials. Bi-coherent states of the Gazeau-Klauder type are constructed and their properties are analyzed. Some examples are also discussed, including an application to the Black-Scholes equation, one of the most important equations in Finance.
Highlights
Supersymmetric quantum mechanics (Susy qm, in the following) is nowadays a well analyzed approach which has proven to be quite useful in the attempt of constructing
It is useful to draw the following picture: This diagram shows the effects of SUSY and of the adjoint map: SUSY exchanges the order of the operators factorizing the various Hamiltonians mapping the “(1)” into the “(2)” sets of vectors, and viceversa, while keeping unchanged the eigenvalues
Our results look somehow richer, since the adjoint map has interesting features both for its physical consequences and from the mathematical side
Summary
Supersymmetric quantum mechanics (Susy qm, in the following) is nowadays a well analyzed approach which has proven to be quite useful in the attempt of constructing. Hamiltonians whose eigenvalues and eigenvectors can be deduced, out of those of a given operator. The role of factorization in this procedure is crucial, and it is widely discussed. We refer to [1,2,3] for many results on Susy qm and to [4] for an interesting review on the factorization method, with a very reach list of references. W(x), at least if w(x) is a real function, called superpotential. The domains of a and a†, D(a) and D(a†), cannot be all of H, since each function in these sets must be, at least, differentiable
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