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Smeared quantum lattices exhibiting \(\mathscr{PT}\)-symmetry with positive \(\mathscr{P}\). (English) Zbl 1338.81426

Summary: A new strategy of the use of \(\mathscr{PT}\) symmetry in quantum theory is proposed. The essence of the innovation lies in the replacement of the usual parity-like choice of \(\mathscr{P}\) by its non-involutory and positive-definite alternative \(\mathscr{P}^{(\mathrm{positive})} \neq I\). The resulting modified concept of \(\mathscr{P}^{(\mathrm{positive})}\mathscr{T}\)-symmetry remains phenomenologically appealing as well as technically useful. This is demonstrated and illustrated via an \(N\)-site quantum lattice model which is exactly solvable in terms of Legendre polynomials.

MSC:

81V55 Molecular physics
82D25 Statistical mechanics of crystals
81V70 Many-body theory; quantum Hall effect
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
81Q80 Special quantum systems, such as solvable systems
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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