Spectral Tur\'an-type problems for the $\alpha$-spectral radius of hypergraphs with degree stability

An $r$-pattern $P$ is defined as an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be $P$-colorable if there is a homomorphism $\phi$: $V(H)\rightarrow [l]$ such that the $r$-multiset $\{\phi(v_{1}),\ldots,\phi(v_{r})\}$ is in $E$ for every edge $\{v_{1},\ldots,v_{r}\}\in E(H)$. Let $Col(P)$ denote the family of all $P$-colorable $r$-graphs. This paper establishes spectral extremal results for $\alpha$-spectral radius of hypergraphs using analytic techniques. We show that for any family $\mathcal{F}$ of $r$-graphs that is degree-stable with respect to $Col(P)$, spectral Tur\'an-type problems can be effectively reduced to spectral extremal problems within $Col(P)$. As an application, we determine the maximum $\alpha$-spectral radius ($\alpha\geq1$) among all $n$-vertex $F^{(r)}$-free $r$-graphs, where $F^{(r)}$ represents the $r$-expansion of the color critical graph $F$. We also characterize the corresponding extremal hypergraphs. Furthermore, leveraging the spectral method, we derive a corresponding edge Tur\'an extremal result. More precisely, we show that if $\mathcal{F}$ is degree-stable with respect to $Col(P)$, then every $\mathcal{F}$-free edge extremal hypergraph must be a $P$-colorable hypergraph.