  We study the $L^2$-gradient flow of functionals $\mathcal F$ depending on the
eigenvalues of Schr\"odinger potentials $V$ for a wide class of differential
operators associated to closed, symmetric, and coercive bilinear forms,
including the case of all the Dirichlet forms (as for second order elliptic
operators in Euclidean domains or Riemannian manifolds).
  We suppose that $\mathcal F$ arises as the sum of a $-\theta$-convex
functional $\mathcal K$ with proper domain $\mathbb{K}\subset L^2$ forcing the
admissible potentials to stay above a constant $V_{\rm min}$ and a term
$\mathcal H(V)=\varphi(\lambda_1(V),\cdots,\lambda_J(V))$ which depends on the
first $J$ eigenvalues associated to $V$ through a $C^1$ function $\varphi$.
  Even if $\mathcal H$ is not a smooth perturbation of a convex functional (and
it is in fact concave in simple important cases as the sum of the first $J$
eigenvalues) and we do not assume any compactness of the sublevels of $\mathcal
K$, we prove the convergence of the Minimizing Movement method to a solution
$V\in H^1(0,T;L^2)$ of the differential inclusion $V'(t)\in
-\partial_L^-\mathcal F(V(t))$, which under suitable compatibility conditions
on $\varphi$ can be written as \[
V'(t)+\sum_{i=1}^J\partial_i\varphi(\lambda_1(V(t)),\dots,
\lambda_J(V(t)))u_i^2(t)\in -\partial_F^-\mathcal K(V(t)) \] where
$(u_1(t),\dots, u_J(t))$ is an orthonormal system of eigenfunctions associated
to the eigenvalues $(\lambda_1(V(t)), ,\dots,\lambda_J(V(t)))$.
