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arXiv · 2201.04526

An Exact Perturbative Existence and Uniqueness Theorem

Nikita Nikolaev
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Abstract

We investigate singularly perturbed nonlinear complex differential systems of the form $\hbar \partial_x f = F (x, \hbar, f)$ where $\hbar$ is a small complex perturbation parameter. Under a geometric assumption on the eigenvalues of the Jacobian matrix of $F$, we prove an Existence and Uniqueness Theorem for exact perturbative solutions; i.e., holomorphic solutions with prescribed perturbative expansions in $\hbar$. In fact, these solutions are the Borel resummation of the formal perturbative solutions.

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Nikita Nikolaev. 2022-01-12. An Exact Perturbative Existence and Uniqueness Theorem. https://arxiv.org/abs/2201.04526

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