On the existence of eigenvalues of a divergence-form operator A+λB in a gap of σ(A)

We consider uniformly elliptic divergence type operators A=−Σ∂_ja_ij(x)∂_i with bounded, Lipschitz continuous coefficients, acting in the Hilbert space L₂(R^ν). It is easy to see that such an operator cannot have discrete eigenvalues below the infimum of the essential spectrum. In order to produce eigenvalues with exponentially decaying eigenfunctions we study the family of operators A+λB=−Σ∂_j(a_ij(x)+λb_ij(x))∂_i, λ≥0, where A is supposed to have a spectral gap, while (b_ij)≥0 and b_ij(x)→0, as x→∞. One of our main results assures that discrete eigenvalues of A+λB move into the gap, as λ increases, if the support of the matrix function (b_ij) is large enough. In addition, we analyze the connection between decay properties of the coefficient matrix (b_ij) and the asymptotics of the associated eigenvalue counting function; these results are modeled on our earlier work in the Schrödinger case.