Orthogonality to a one parameter family of eigenfunctions

Question

Let $\rho>0$ be a smooth realvalued function such that $\rho=1$ outside the unit interval $(-1,1)$. For each $t>0$, let us denote by $\{\lambda_n(t)\}_{n=1}^{\infty}$ and $\{\phi_n(t;x)\}_{n=1}^{\infty}$ the eigenvalues (written in strictly increasing order) and eigenfunctions associated to the equation $$ -\partial^2_x \phi_n(t;x) = \lambda_n(t) \rho(x) \phi_n(t;x) \quad \text{on $(-t,t)$},$$ subject to Dirichlet boundary condition $\phi_n(t;\pm t)=0$.

Suppose that $f\in C^{\infty}_0((-1,1))$ and that $$ \int_{-1}^{1} f(x) \phi_n(t;x) \,dx =0 $$for some $n \in \mathbb N$ and all $t>1$. Does it follow that $f=0$?

Answer 1

The answer is no.

Here is a counterexample: take $\rho(x)=1$ for all $x$ and choose the eigenfunction $\phi_1(t;x)=\cos(\pi x/2t)$; then the integral $\int_{-1}^{1} f(x) \phi_1(t;x) \,dx =0$ for all odd functions $f(x)$, irrespective of $t>0$.