Fredholm determinant and the Sturm-Liouville problems in quantum mechanics

Anargyros Papageorgiou, H. Woźniakowski

João Philipe Macedo Braga, R. N. Costa Filho

The modified Schrödinger equation obtained by Costa Filho et al. [Phys. Rev. A 84, 050102(R) (2011)] is shown to be a Sturm–Liouville problem. This demonstration guarantees that Hamiltonian eigenvalues obtained in this formalism are real. It also allows us to show that, regardless of the non-Hermitian characteristic of the Hamiltonian operator in the Hilbert space, its time evolution remains unitary.

Carl M. Bender, Hugh F Jones

Most studies of -symmetric quantum-mechanical Hamiltonians have considered the Schrödinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm–Liouville problems, the eigenvalues of the -symmetric Sturm–Liouville problem grow like n 2 for large n . However, the novelty is that a eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Martin Wilner