On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems

In the present paper, we are concerned with the Sturm–Liouville operator
$$\mathcal{L}[q] u:=-u''+q(x)u$$
subject to the separated boundary conditions. We suppose that $q \in L^2(0,\pi)$ and study a so-called inverse optimization spectral problem: given a potential $q_0$ and a value $\lambda_k $, where $k=1,2,\dots$, find a potential $\hat{q}$ closest to $q_0$ in the norm of $L^2(0,\pi)$ such that the value $\lambda_k$ coincides with $k$-th eigenvalue $\lambda_k(\hat{q})$ of the operator $\mathcal{L}[\hat{q}]$.
In the main result, we prove that this problem is related to the existence of a solution to a boundary value problem for the nonlinear equation
$$ -u''+q_0(x) u=\lambda_k u+\sigma u^3 $$
with $\sigma=1$ or $\sigma=-1$. This implies that the minimizing solution of the inverse optimization spectral problem can be obtained by solving the corresponding nonlinear boundary value problem. On the other hand, this relationship allows us to establish an explicit formula for the solution to the nonlinear equation by finding the minimizer of the corresponding inverse optimization spectral problem. As a consequence of this result, a new method of proving the generalized Sturm nodal theorem for the nonlinear boundary value problems is obtained.