## 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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