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EDIT: I describe the results which seem most relevant to your question:

Let $t\mapsto A(t)$ for $t\in T$ be a parameterized family of unbounded operators in a Hilbert space $H$ with common domain of definition and with compact resolvent.

If $t\in T=\mathbb R$ and all $A(t)$ are self-adjoint then the following holds:

(A) (Rellich) If $A(t)$ is real analytic in $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized real analytically in $t$.

(B) If $A(t)$ is quasianalytic of class $C^Q$ in $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized $C^Q$ in $t$.

If $t\in T=\mathbb R^n$ and all $A(t)$ are normal then the following holds:

(L) If $A(t)$ is real analytic or quasianalytic of class $C^Q$ in $t\in \mathbb R^n$, then for each $t_0\in \mathbb R^n$ and for each eigenvalue $z_0$ of $A(t_0)$, there exist a neighborhood $D$ of $z_0$ in $\mathbb C$, a neighborhood $W$ of $t_0$ in $\mathbb R^n$, and a finite covering ${\pi_k : U_k \to W}$ of $W$, where each $\pi_k$ is a composite of finitely many mappings each of which is either a local blow-up along a real analytic or $C^Q$ submanifold or a local power substitution, such that the eigenvalues of $A(\pi_k(s))$, $s \in U_k$, in $D$ and the corresponding eigenvectors can be parameterized real analytically or $C^Q$ in $s$. If $A$ is self-adjoint, then we do not need power substitutions.

(M) If $A(t)$ is real analytic or quasianalytic of class $C^Q$ in $t\in \mathbb R^n$, then for each $t_0\in \mathbb R^n$ and for each eigenvalue $z_0$ of $A(t_0)$, there exist a neighborhood $D$ of $z_0$ in $\mathbb C$ and a neighborhood $W$ of $t_0$ in $\mathbb R^n$ such that the eigenvalues of $A(t)$, $t \in W$, in $D$ and the corresponding eigenvectors can be parameterized by functions which are special functions of bounded variation (SBV) in $t$.

In particular, if your mapping $f$ is Hölder continuous (of class $C^{0,\alpha}$ for $0<\alpha\le1$) then the eigenvalues can be parameterized in a $C^{0,\alpha}$ way also.