We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from -symmetric quantum mechanics.
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Recall that “self-adjointness” of a densely defined operator H, i.e., H = H∗, is a much stronger property than being just “Hermitian” or “symmetric,” i.e., H ⊂ H∗. On the other hand, the terms “non-Hermitian” and “non-self-adjoint” can be read as synonyms in the present paper, since we are concerned with a “fundamental non-self-adjointness” caused, for instance, by complex coefficients of differential operators, that is, we are not interested in closed symmetric non-self-adjoint operators here.
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