Bootstrapping PT symmetric Hamiltonians

Abstract

Bootstrapping in Quantum Mechanics uses positivity condition to derive the Eigen spectum. For non-hermitian systems usual positivity condition does not work. In this paper we define positivity condition for special class of non-hermitian hamiltonian, the PT symmetric Hamiltonian. We illustrate this modified positivity condition with several examples and obtain eigen spectrum.

The Intellectual Legacy of Sun Yat-sen

“…The explicit expressions of G + and G − are17 It is precisely the nontrivial dependence on H that leads to the nonlinear energy spacing in the occupation number n. Furthermore, G + and G − are not independent as they are closely related to the energy differences. If the operator H in G ± is replaced by a number E, we have…”

“…17 Notice that G+ = −G−|H→−H,g→−g + O(g 4 ). For the harmonic oscillator, the transformations x → ix, p → ip, i.e.…”