A naive, sophomore-level application of the elementary first-order
perturbation formula to the equally elementary ODE Eqs. (13) and
(18). The introductory reference to non-commutative geometry is even
more naive, not paying, i.a.,  any attention to the existence of the
(incidentally, strongly singular) Schroedinger operators in
question.

MR2821745 Motavalli, Hossein; Rezaei Akbarieh, Amin Generalized
spiked harmonic oscillator in non-commutative space. Internat. J.
Theoret. Phys. 50 (2011), no. 9, 2673--2678. 81Rxx