Compact and neat presentation of a fresh idea -- either a proposal
of a new family of Schroedinger equations for quantum theory or a
proposal of a new realization of Hilbert spaces. For accelerated
reading I would recommend skipping all the tutorials and start
reading from the last section 6 which looks like just a summary of
the standard principles of quantum mechanics. Using a slightly
strange notation. The impatient reader is then recommended to
search for definitions in section 5. There, the notation is
explained as, in essence, converting the vector space of wave
functions into the Hilbert space of states. Based on the
introduction of a certain fairly specific and nonstandard inner
product called q-deformed product. Now, the reader understands the
point and may return, if necessary, to section 2 (which explains
what is q-calculus and how one can get the Jackson's
representation of q-deformed Heisenberg algebra) and sections 3
and 4 which recollect some author's older results on q-deformed
Fokker-Planck equation which, via  the Risken's stochastic
quantization, enable him to arrive finally at the q-deformed
Hamiltonian (34).

MR2455812 Lavagno, A. Deformed quantum mechanics and $q$-Hermitian
operators. J. Phys. A: Math. Theor. 41 (2008), no. 24, 244014, 9
pp. 81S05 (39A20 81Q05 81R50)