Spectral Asymptotics for \(\mathcal {P}\mathcal {T}\) Symmetric Operators

  • Johannes Sjöstrand
Part of the Pseudo-Differential Operators book series (PDO, volume 14)


\(\mathcal {P}\mathcal {T}\)-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 Schrödinger 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
$$\displaystyle [P,\mathcal {P}_\iota \mathcal {T} ]=0. $$


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Authors and Affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-ComtéDijonFrance

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