## Non-Hermitian quantum mechanics.(English)Zbl 1230.81004

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.

### 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
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