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Please , can someone help me to understand this part of text please

What it means "... for some $k\in \mathbb{N} \cup \lbrace 0 \rbrace$ and uniformly fore a.e $t\in [0,2\pi]$"

and please how to find the eigenvalues $\lambda=m^2$ of $(P_0)$ .

please what it means :"depends on interplay of the behaviors of f near infinity and near zero with the eigenvalues of $(P_0)$"

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Please , Thank you

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