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Spectral Asymptotics for $$\mathcal {P}\mathcal {T}$$ Symmetric Operators

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\(\mathcal {P}\mathcal {T}\)-symmetry has been proposed as an alternative to self-adjointness in quantum physics, see Bender et al. (J Math Phys 40(5):2201–2229, 1999), Bender and Mannheim (Phys Lett A 374(15–16):1616–1620, 2010). Thus for instance, if we consider a Schrodinger operator on Rn, $$\displaystyle P=-h^2\Delta +V(x), $$ the usual assumption of self-adjointness (implying that the potential V is real valued) can be replaced by that of \(\mathcal {P}\mathcal {T}\)-symmetry: $$\displaystyle V\circ \iota =\overline {V}, $$ where ι : Rn →Rn is an isometry with ι2 = 1≠ι. If we introduce the parity operator \(\mathcal {P}_\iota u(x)=u(\iota (x))\) and the time reversal operator \(\mathcal {T} u=\overline {u}\), then this can be written $$\displaystyle [P,\mathcal {P}_\iota \mathcal {T} ]=0. $$