(Context:) I was reading Andrew Hassel's article: Ergodic billiards that are not quantum unique ergodic (see page 4) and I got puzzled by his paraphrasing of Heller & O'Connor and Zelditch argument why a certain billiard stadium is non-QUE.
Namely, consider rectangle of height $h = \pi$ and width $w = th$ for $t \in [1, 2]$. Call the region formed by joining the said rectangle with two semicircles to its sides with centres at $(\pm t\pi/2, 0)$ and radii $\frac{\pi}{2}$. Let $f_n$ be given by $v_n = \chi(x)\sin(ny)$ for even $n$ and $v_n = \chi(x)\cos(ny)$ for odd $n$ where $\chi$ is supported in $x \in [-\pi/4, \pi/4]$ and is chosen such that $||v_n||_{L^2} = 1$.
Hassell states that
They [the $v_n$s] satisfy $||(\Delta - n^2)v_n||_{L^2}\leq K$, uniformly in $n$. It follows from basic spectral theory that $$\frac{3}{4}\leq ||P_{[n^2 - 2K, n^2 + 2K]}v_n||^2$$ where $P_I$ is the spectral projection operator of $\Delta$ corresponding to the set $I\subset \mathbb{R}$.
In the rest of his paraphrasing, he essentially states that if the number of eigenvalues of $\Delta$ is bounded by a uniform constant $M$ in the intervals $[n_j^2 - 2K, n_j^2 + 2K]$, then there exists at least one orthonormal eigenfunction of $\Delta$ with an eigenvalue in the range $[n_j^2 - 2K, n_j^2 + 2K]$ due to $\frac{3}{4}\leq ||P_{[n^2 - 2K, n^2 + 2K]}v_n||^2$.
(My question:) My intuition about the argument why such an eigenfunction exists that by the inequality, the function $v_n$ is at least $\frac{3}{4}$ an eigenfunction with an eigenvalue in the projected interval. Therefore there must exist some suitable eigenfunction with an eigenvalue in the appropriate range, for otherwise the projection would be equal to zero - so in fact what only matters is that the lb. is non-zero!
But what I am unsure about is 1.) why exactly does it follow from "basic spectral theory and 2.) how is the orthogonal projection operator formulated in the case of eigenfunctions of $\Delta$? I am asking for either a straight answer or a reference which clarifies these (and specifically these) questions. Thanks!