A remark on the Dirichlet-Neumann decoupling and the integrated density of states (Q5927509)

An estimate on the difference of the number of eigenvalues for Schrödinger operators with Dirichlet and Neumann boundary conditions in large boxes is obtained. The Schrödinger operator has the form NEWLINE\[NEWLINE H=(p-A(x))^2+V(x) \qquad \text{ on } L^2(\mathbb{R}^d) NEWLINE\]NEWLINE with \(d\geq 1\), where \(p=-i\partial_x\) is the momentum operator, \(A(x)\) is a vector potential and \(V(x)\) is a scalar potential. The magnetic field is given by NEWLINE\[NEWLINE B_{ij}(x)=\partial_{x_i}A_j(x)-\partial_{x_j}A_i(x), \qquad i\neq j, \quad x\in \mathbb{R}^d. NEWLINE\]NEWLINE The proof of the main result is based on Krein's theory of a spectral shift function. The general theory is used for studying the integrated density of states. It is shown that the integrated density of states is independent of the boundary conditions.