Stability of the problem of recovering the Sturm–Liouville operator from the spectral function

We consider a differential operator $\mathscr L=(h,q(x))$ generated by a Sturm-Liouville operation $l[y]=-y''+q(x)y$ on the linear manifold of finite twice-differentiable functions $y(x)$ satisfying the boundary condition $y'(0)-hy(0)=0$. Let $\rho(\mu)$ be the spectral function of this operator. From $\rho(\mu)$, as is well known, we can recover the operator $\mathscr L$, i.e. the number $h$ and the function $q(x)$. Let $V_\alpha^A$ be the set of operators $\mathscr L$ for which
$$ |h|\leqslant A,\qquad\int_0^x|q(t)|\,dt\leqslant\alpha(x)\quad(x<0<\infty). $$

We now investigate how much information about the operator $\mathscr L\in V_\alpha^A$ can be obtained if its spectral function $\rho(\mu)$ is known only for values of $\mu$ on a finite interval.
In the present article we obtain estimates for the difference in the potentials $q_1(x)-q_2(x)$, in the boundary parameters $h_1-h_2$ and in the solutions of the corresponding differential equations under the condition that the spectral functions of the two operators in $V_\alpha^A$ coincide on a finite interval.
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