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$$\mathcal P\mathcal T$$-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 Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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##### References:
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