I am trying to understand this short paper and I am getting stuck right at the end.
Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$). We are considering the operator
on $L^2([0,1],dx)$ where $A$ has periodic boundary condition $f'(1)=f'(0)$, $f(1)=f(0)$.
Suppose that for some $n$ even, the $n+1$st eigenvalue of $A$ coincides with the $n$th eigenvalue of $A$. We call that eigenvalue $\hat E$. In other words, there is a multiplicity 2 eigenvalue of $A$ at $\hat E$.
It is known that all solutions $u\in L^2 (\mathbb R, dx)$ to the Schrödinger equation $$-u''(x)+V(x)u(x)=\hat E u(x)$$ are periodic. Let $u_1(x)$ be the solution with initial conditions $u(0)=0, u'(0)=1$ and let $u_2(x)$ be the solution with initial conditions $u(0)=1, u'(0)=0$. It is easy to find a 1-periodic function $W(x)$ in $C^\infty$ that satisfies
$$\int_0^1 W(x) u_1(x)^2 dx\neq \int_0^1 W(x) u_2(x)^2 dx.$$
Let $\lambda$ be a small constant. Supposedly, when we perturb $V$ by $\lambda W$ in the definition of $A$ (that is, replace $V$ with $V+\lambda W$) we can guarantee that for sufficiently small $\lambda$ the $n+1$st eigenvalue of $A$ no longer coincides with the $n$th eigenvalue. Why is that true?
The paper cites Kato's Perturbation Theory for Linear Operators as justification, but I cannot find anything there that is directly relevant.