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An array of N subsequent Laguerre polynomials is interpreted as aneigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrumor, better said, of an exactly solvable N-site-lattice cryptohermitianHamiltonian whose spectrum is known as equal to the set of zeros of the N-thLaguerre polynomial. The two key problems (viz., the one of the ambiguity andthe one of the closed-form construction of all of the eligible inner productswhich make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) arediscussed. Then, for illustration, the first four simplest, $k-$parametricdefinitions of inner products with $k=0,k=1,k=2$ and $k=3$ are explicitlydisplayed. In mathematical terms these alternative inner products may beperceived as alternative Hermitian conjugations of the initial N-plet ofLaguerre polynomials. In physical terms the parameter $k$ may be interpreted asa measure of the "smearing of the lattice coordinates" in the model.

An exactly solvable quantum-lattice model with a tunable degree of nonlocality