Let $(A_\epsilon)_{\epsilon}$ be a family of self-adjoint and below-bounded operators in the Hilbert space $H$. Assume that the essential spectrum $\sigma_{ess}(A_\epsilon)$ is the same for all $A_\epsilon$. Moreover, we assume that $A_\epsilon$ is depends monotonically on $\epsilon$ (the inequality $A_{\epsilon_1}\leq A_{\epsilon_2}$ is understood in the form sense).
Let $(a,b)\cap\sigma_{ess}(A_\epsilon)=\emptyset$, and let for each $\epsilon$ the spectrum of $A_\epsilon$ within $(a,b)$ consists of one simple eigenvalues; we denote this eigenvalue $\lambda_\epsilon$.
Question: is it true that the function $\epsilon\mapsto \lambda_\varepsilon$ is monotonic? And if yes, how to prove that?
If $(a,b)$ lies below the infimum of the essential spectrum, the answer is indeed YES -- the proof is simple (apply min-max principle). But what to with $(a,b)$ lying above this infimum?