Abstract
Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure generated by unbounded metric operators in a Hilbert space. To that effect, we consider the notions of similarity and quasi-similarity between operators and explore to what extent they preserve spectral properties. Then we study quasi-Hermitian operators, bounded or not, that is, operators that are quasi-similar to their adjoint and we discuss their application in pseudo-Hermitian quantum mechanics. Finally, we extend the analysis to operators in a partial inner product space (pip-space), in particular the scale of Hilbert space s generated by a single unbounded metric operator.
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- 1.
Non-self-adjoint resolutions of the identity have recently be studied by Inoue and Trapani [17].
- 2.
Self-adjoint operators are usually called Hermitian by physicists.
- 3.
The space \({\mathscr {H}}(R_G^{-1})\) is (three times) erroneously denoted \({\mathscr {H}}(R_{G^{-1}})\) in [2, p. 4]; see Corrigendum.
- 4.
Since G is self-adjoint, the expression \(G > 1\) makes sense as a shortcut for \(\langle {G\xi } |{\xi } \rangle > \left\| \xi \right\| _{}^2, \forall \, \xi \in D(G).\)
- 5.
- 6.
KLMN stands for Kato, Lax, Lions, Milgram, Nelson.
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Antoine, JP., Trapani, C. (2016). Operator (Quasi-)Similarity, Quasi-Hermitian Operators and All that. In: Bagarello, F., Passante, R., Trapani, C. (eds) Non-Hermitian Hamiltonians in Quantum Physics. Springer Proceedings in Physics, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-31356-6_4
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DOI: https://doi.org/10.1007/978-3-319-31356-6_4
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