The paper under review deals with trace identities concerning the eigenvalues \(\lambda_j\) and the eigenvectors \(\phi_j\) of a self-adjoint operator \(H\) acting in a Hilbert space. In a first main theorem, it is proved that \[ \sum_k \frac{| \langle [H,G]\phi_j,\phi_k\rangle| ^2 }{ \lambda_k-\lambda_j} = -\frac{1}{ 2} \langle [[H,G],G]\phi_j,\phi_j\rangle= \sum_k (\lambda_k-\lambda_j)| \langle G\phi_j,\phi_k\rangle| ^2 \] if \(G\) is a second selfadjoint operator such that \(G(D_H)\subset D_H\).
The second main result is a generalization to the case of two operators, \(H_1\) and \(H_2\) (\(H_2\) not necessarily selfadjoint) with the mixing commutator \(H_1G-GH_2\) instead of \([H,G]\), the model case being Laplacians with different boundary conditions. This second theorem contains an estimate of the eigenvalues of \(H_1\) in terms of those of \(H_2\).
The authors include several interesting applications when \(H\) is a positive elliptic operator with Dirichlet boundary conditions, the Dirichlet Laplacian, the Neumann Laplacian, and when \(H_1\) and \(H_2\) are a couple of Schrödinger operators. The second operator \(G\) is the multiplication by a function or, in the last case, the derivative.