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Abstract

Let $\ell_j:=-{d^2}/{dx^2}+k^2q_j(x),$ $k={\rm const}>0$, $j=1,2,$ $0< \mathop{\rm ess\,inf} q_j(x)\leq \mathop{\rm ess\,sup} q_j(x)<\infty.$ Suppose that $(*)\kern.5em \int_0^1p(x)u_1(x,k)u_2(x,k)\,dx=0$ for all $k>0,$ where $p$ is an arbitrary fixed bounded piecewise-analytic function on $[0,1]$, which changes sign finitely many times, and $u_j$ solves the problem $\ell_ju_j=0,\ 0\leq x\leq 1,\ u'_j(0,k)=0,\ u_j(0,k)=1.$ It is proved that $(*)$ implies $p=0$. This result is applied to an inverse problem for a heat equation.