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Complex Square Well --- A New Exactly Solvable Quantum Mechanical Model

Quantum Physics 2008-11-26 v1 Condensed Matter High Energy Physics - Theory Mathematical Physics math.MP

Abstract

Recently, a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian H=p2+x2(ix)ϵH=p^2+x^2(ix)^\epsilon was studied. It was found that the energy levels for this theory are real for all ϵ0\epsilon\geq0. Here, the limit as ϵ\epsilon\to\infty is examined. It is shown that in this limit, the theory becomes exactly solvable. A generalization of this Hamiltonian, H=p2+x2M(ix)ϵH=p^2+x^{2M}(ix)^\epsilon (M=1,2,3,...) is also studied, and this PT-symmetric Hamiltonian becomes exactly solvable in the large-\epsilon limit as well. In effect, what is obtained in each case is a complex analog of the Hamiltonian for the square well potential. Expansions about the large-\epsilon limit are obtained.

Keywords

Cite

@article{arxiv.quant-ph/9906057,
  title  = {Complex Square Well --- A New Exactly Solvable Quantum Mechanical Model},
  author = {Carl M. Bender and Stefan Boettcher and H. F. Jones and Van M. Savage},
  journal= {arXiv preprint arXiv:quant-ph/9906057},
  year   = {2008}
}

Comments

7 pages, Revtex, 2 eps-figures enclosed