Non-Hermitian quantum mechanics. (English) Zbl 1230.81004

Moiseyev, Nimrod

Cambridge: Cambridge University Press (ISBN 978-0-521-88972-8/hbk; 978-0-511-98557-7/ebook). xiii, 394 p. (2011).

The author shows the theory of the non-Hermitian quantum mechanics (NHQM) in a form with exercises and answers to them. In Exercise 1.4, he treats the Hamiltonian \(H= H_0+\lambda V\) satisfying \(H_0(x)= H_0(-x)\) and \(V(x)= -V(-x)\). When \(\lambda=i\Gamma\) and \(\Gamma\) is real-valued only, \(H\) commutes with the \(PT\) symmetry operator: \(PxP^{-1}= -x\) and \(TiT^{-1}= -i\). The eigenvalues of \(H(x,\Gamma)\) are real for \(|\Gamma|< |\lambda_{bp}|\). As an answer, let \((H_0+ \lambda V)\psi_j= E_j\psi\). A perturbative expansion of \(\psi_j\) and \(E_j\) in a power series of \(\lambda\) converge for \(|\lambda|<|\lambda_{bp}|\). For \(\lambda= i\Gamma\) and \(\lambda_{bp}= i\Gamma_{bp}\), we can write \[ E(\lambda)= E^{bp}\pm D\{(\lambda- \lambda_{bp})(\lambda- \lambda^*_{bp})\}^{1/2} \] in the vicinity of \(\lambda_{bp}\), as the most common situation: \[ E_j(|\lambda|< |\lambda_{bp}|)= \sum_{n=0\sim\infty} \lambda^n E^{(n)}_j;\;E_j= \langle\chi^{(n)}_j|H_0+\lambda V|\chi^{(n)}_j\rangle+ 0(\lambda^{2n+2}), \]
\[ \chi^{(n)}_j(x)= \sum_{k=0\sim n}\lambda^k\psi^{(k)}_j,\;E^{(2n)}_j= \langle\psi^{(n)}_j|V|\psi^{(n-1)}_j\rangle. \] Since \(E^{(2n+1)}_j= 0\), \(n= 0,1,\dots\), are derived, \[ E_j(|\Gamma|<|\Gamma_{bp}|)= \sum_{n=0\sim\infty}(-1)^n \Gamma^{2n}E^{(2n)}_j, \] has real values only.
The author also shows in Fig.1.1 the metastable resonance states for a particle in a spherically symmetric potential barrier given by \(V(r)= (r^2/2-0.8)\exp(-0,1r^2)\). The non-Hermitian methods for the calculation of resonance energies and wave functions are described in Chapters 4–5. As he explains in Chapter 6, the expectation value of a complex scaled dynamical operator \(\widehat 0\) is given by \[ \widetilde 0= \langle\psi^* |\widehat 0|\psi\rangle/\langle\psi^*|\psi\rangle\equiv |\widetilde 0| e^{i\alpha}. \] The physical interpretations of \(\widetilde 0\) are also given. He treats the examples, like to the one in the above Exercise 1.4, in Chapter 9. That is, at the branch point \(\lambda= \lambda_{bp}\), \(\psi_1(r; \lambda_{bp})= \psi_2(r; \lambda_{bp})\equiv\psi_{bp}\) and \((\psi_{bp}|\psi_{bp})=0\) happen. The eigenfunction \(\psi_{bp}(r)\) is referred as a self-orthogonal state, and studied from the standpoint of the NHQM. In the last Chapter 10, the points where QM branches into two formalisms (HQM and NHQM) are discussed.

Reviewer: Hideo Yamagata (Osaka)

MSC:

81-02	Research exposition (monographs, survey articles) pertaining to quantum theory
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q12	Nonselfadjoint operator theory in quantum theory including creation and destruction operators