MR2062422 J. Form\'anek, R. J. Lombard and J. Mare\v s "Wave
equations with energy-dependent potentials" Czechoslovak J. Phys.
54 (2004), no. 3, 289--316

In quantum phenomenology, Schr\"odinger equations are often (and
often vaguely) treated as representing so-called "effective"
theories. One of the manifestly consistent and quantitative
implementations of the latter idea is that one simply projects the
original "exact" equation on a suitable "model" subspace of
Hilbert space. In such (and similar) cases, the generator of time
evolution (i.e., the "Hamiltonian" or "energy operator" $H$)
becomes "nonlinear" (in the sense that its form varies with the
varying-quantized-energy levels).

The authors attack such a nonlinearity problem in a way motivated
by physics (questions of interpretation of similar systems are
briefly discussed and amply---if not overly much---illustrated). A
few related contradictory topics (e.g., the concepts of the
"correct" scalar product or of the time-evolution in similar
systems) are disentangled.

Marginally, let me note that one of the most challenging questions
opened in this paper concerns our appropriate understanding of the
meaning of the non-Hermiticity of $H=H(E)$. I must admit that in
this sense this paper inspired my own recent analysis \ref[Phys.
Lett. A 326 (2004), no. 1-2, 70--76 MR2065888; Czechoslovak J.
Phys. 54 (2004), no. 10, 1143--1148] so that I can recommend
Form\'anek, Lombard and Mare\v s' paper as truly inspiring.