Susy for Non-Hermitian Hamiltonians, with a View to Coherent States
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Susy for Non-Hermitian Hamiltonians, with a View to Coherent States F. Bagarello1,2 Received: 13 March 2020 / Accepted: 6 July 2020 / © The Author(s) 2020
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. Keywords Supersymmetic quantum mechanics · Ladder operators · Non self-adjoint hamiltonian · Gazeau-Klauder coherent states Mathematics Subject Classification (2010) 81Q60 · 81Sxx
1 Introduction 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 easily 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 [1–3] for many results on Susy qm and to [4] for an interesting review on the factorization method, with a very reach list of references. d The essence is the following: we consider an operator a = dx + w(x), acting on d 2 † H ≡ L (R), whose adjoint is a = − dx + w(x), at least if w(x) is a real function, called superpotential. Needless to say, 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. F. Bagarello
[email protected] 1
Dipartimento di Ingegneria, Universit`a di Palermo, 90128 Palermo, Italy
2
INFN, Sezione di Napoli, Napoli, Italy
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Math Phys Anal Geom
(2020) 23:28
This suggests that, in general, they are unbounded, since all closed bounded operators can be defined everywhere in H. For instance, if we take w(x) linear in x as for harmonic oscillator, it is well known that a and a † are unbounded. However, all throughout this paper, we will not consider in details this aspect of the operators involved in our analysis, except when it will be essential. Two operators can now be introduced: h1 = a † a and h2 = aa † . In the coordinate representation, these look like: h1 = a † a = −
d2 + v1 (x), dx 2
h2 = aa † = −
d2 + v2 (x), dx 2
(1.1)
where v1 (x) = w2 (x) − w (x),
v2 (x) = w2 (x) + w (x).
(1.2)
It is easy to check that [a, a † ] = h2 − h1 = 2w (x), which is zero only if the superpotential is constant. Notice that h1 and h2 are both Hermitian and non-negative: f, hj f 0, j = 1, 2, for all f ∈ D(hj ), the domain of hj . Hence all their eigenvalues are real and non-negative. It is clear that the two vacua of a and a † cannot be both square-integrable. In fact, assuming that ϕ (1) (x) and ϕ (2) (x) satisfy aϕ (1) (x) = 0 and a † ϕ (2) (x) = 0, we find that (1) (2) ϕ (x) = N1 exp − w(x) ,
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