##
**\(\mathcal P\mathcal T\)-symmetric quantum mechanics.**
*(English)*
Zbl 1057.81512

Summary: This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition \(H^\dagger=H\) on the Hamiltonian, where \(\dagger\) represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian \(H\) has a real spectrum. However, replacing this mahematical condition by the weaker and more physical requirement \(H^‡=H\), where \(‡\) represents combined parity reflection and time reversal \(\mathcal P\mathcal T\), one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation \(H=p^2+x^2(ix)^\epsilon\) of the harmonic oscillator Hamiltonian, where \(\epsilon\) is a real parameter. The system exhibits two phases: When \(\epsilon\geq 0\), the energy spectrum of \(H\) is real and positive as a consequence of \(\mathcal P\mathcal T\) symmetry. However, when \(-1<\epsilon<0\), the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because \(\mathcal P\mathcal T\) symmetry is spontaneously broken. The phase transition that occurs at \(\epsilon=0\) manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians \(H=p^2+x^{2N}(ix)^\epsilon\) with \(N\) integer and \(\epsilon>-N\); each of these complex Hamiltonians exhibits a phase transition at \(\epsilon=0\). These \(\mathcal P\mathcal T\)-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.

### MSC:

81Q05 | Closed and approximate solutions to the SchrĂ¶dinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

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\textit{C. M. Bender} et al., J. Math. Phys. 40, No. 5, 2201--2229 (1999; Zbl 1057.81512)

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