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

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

## Abstract

$$\mathcal {P}\mathcal {T}$$-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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