On the existence of meromorphic solutions of the complex Schrödinger equation with a q-shift
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| Format: | Preprint |
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2026
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| _version_ | 1866917190588956672 |
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| author | Korhonen, Risto Liu, Wenlong |
| author_facet | Korhonen, Risto Liu, Wenlong |
| contents | In this paper, we study the following complex Schrödinger equation with a $q$-difference term: \begin{align}\tag{†}\label{dagger} f'(z) = a(z)f(qz) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \end{align} where $a(z) \not\equiv 0$ is a small meromorphic function with respect to $f(z)$, and all the coefficient functions of $R(z, f(z))$ are also small meromorphic functions with respect to $f(z)$. We assume that $q\in\mathbb{C}\setminus \left \{ 0,-1,1 \right \} $ and that $R(z, f(z))$ is an irreducible rational function in both $f(z)$ and $z$. We obtain some necessary conditions for \eqref{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 \eqref{dagger}. Additionally, we consider the possible forms of the meromorphic solutions to \eqref{dagger} in certain conditions and examine exponential polynomials as possible solutions of \eqref{dagger}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04923 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence of meromorphic solutions of the complex Schrödinger equation with a q-shift Korhonen, Risto Liu, Wenlong Complex Variables 30D35 In this paper, we study the following complex Schrödinger equation with a $q$-difference term: \begin{align}\tag{†}\label{dagger} f'(z) = a(z)f(qz) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \end{align} where $a(z) \not\equiv 0$ is a small meromorphic function with respect to $f(z)$, and all the coefficient functions of $R(z, f(z))$ are also small meromorphic functions with respect to $f(z)$. We assume that $q\in\mathbb{C}\setminus \left \{ 0,-1,1 \right \} $ and that $R(z, f(z))$ is an irreducible rational function in both $f(z)$ and $z$. We obtain some necessary conditions for \eqref{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 \eqref{dagger}. Additionally, we consider the possible forms of the meromorphic solutions to \eqref{dagger} in certain conditions and examine exponential polynomials as possible solutions of \eqref{dagger}. |
| title | On the existence of meromorphic solutions of the complex Schrödinger equation with a q-shift |
| topic | Complex Variables 30D35 |
| url | https://arxiv.org/abs/2601.04923 |