An array of N subsequent Laguerre polynomials is interpreted as an
eigenvector of a non-Hermitian tridiagonal Hamiltonian H with real spectrum
or, better said, of an exactly solvable N-site-lattice cryptohermitian
Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th
Laguerre polynomial. The two key problems (viz., the one of the ambiguity and
the one of the closed-form construction of all of the eligible inner products
which make H Hermitian in the respective {\em ad hoc} Hilbert spaces) are
discussed. Then, for illustration, the first four simplest, k−parametric
definitions of inner products with k=0,k=1,k=2 and k=3 are explicitly
displayed. In mathematical terms these alternative inner products may be
perceived as alternative Hermitian conjugations of the initial N-plet of
Laguerre polynomials. In physical terms the parameter k may be interpreted as
a measure of the "smearing of the lattice coordinates" in the model.Comment: 35 p