Suppose that we have a graph $G$, define the hamiltonian $H$ on it as $$Hu(x) = \sum_{y\sim x}u(y).$$ Consider the operator $H+V$ where $V$ multiplies the value $u(x)$ in any vertex by the potential in that vertex. Any minor of $H+V+\lambda$ is a polynomial of $\lambda.$ Is it true that it has only real eigenvalues? Consider the dependence of ONE of these eigenvalues on $V$. What bounds on the derivatives $\frac{\partial \lambda}{\partial V(x)}$ of the minors' eigenvalues are known?
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$\begingroup$ By the eigenvalues of the minor you mean the zeros of this polynomial? $\endgroup$– Christian RemlingCommented Nov 26, 2023 at 1:35
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$\begingroup$ Yes, exactly, the zeros of the polynomial $\endgroup$– Станислав КрымскийCommented Nov 27, 2023 at 13:27
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