In this paper, we study the following complex Schrödinger equation with a q-difference term: align†dagger f'(z) = a(z)f(qz) + R(z, f(z)), R(z, f(z)) = (P(z, f(z))/Q(z, f(z))), align where a(z)≡0 is a small meromorphic function with respect to f(z), and all the coefficient functions of R(z,f(z))
Nearby in the stack
are also small meromorphic functions with respect to
f(z)
. We assume that
q∈C∖{0,−1,1}
and that
R(z,f(z))
is an irreducible rational function in both
f(z)
and
z
. We obtain some necessary conditions for dagger to have meromorphic solutions of zero order and non-constant entire solutions, respectively. In particular, if
R(z,f(z))
reduces to a polynomial in
f(z)
with degree at most 2 and all the coefficients are constant, then under this assumption and without imposing any restrictions on the growth order of
f(z),
we prove the existence of entire solutions in many cases, study their number, and further investigate the local and global meromorphic solutions to dagger. Additionally, we consider the possible forms of the meromorphic solutions to dagger in certain conditions and examine exponential polynomials as possible solutions of dagger.
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