Hadamard-type formulas via the Maslov form

Given a star-shaped bounded Lipschitz domain $\Omega\subset{\mathbb R}^d$, we consider the Schr\"odinger operator $L_{\mathcal G}=-\Delta+V$ on $\Omega$ and its restrictions $L^{\Omega_t}_{\mathcal G}$ on the subdomains $\Omega_t$, $t\in[0,1]$, obtained by shrinking $\Omega$ towards its center. We impose either the Dirichlet or quite general Robin-type boundary conditions determined by a subspace ${\mathcal G}$ of the boundary space $H^{1/2}(\partial\Omega)\times H^{-1/2}(\partial\Omega)$, and assume that the potential is smooth and takes values in the set of symmetric $(N\times N)$ matrices. Two main results are proved: First, for any $t_0\in(0,1]$ we give an asymptotic formula for the eigenvalues $\lambda(t)$ of the operator $L^{\Omega_t}_{\mathcal G}$ as $t\to t_0$ up to quadratic terms, that is, we explicitly compute the first and second $t$-derivatives of the eigenvalues. This includes the case of the eigenvalues with arbitrary multiplicities. Second, we compute the first derivative of the eigenvalues via the (Maslov) crossing form utilized in symplectic topology to define the Arnold-Maslov-Keller index of a path in the set of Lagrangian subspaces of the boundary space. The path is obtained by taking the Dirichlet and Neumann traces of the weak solutions of the eigenvalue problems for $L^{\Omega_t}_{\mathcal G}$.