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 \[ H=(p-A(x))^2+V(x) \qquad \text{ on } L^2(\mathbb{R}^d) \] 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 \[ B_{ij}(x)=\partial_{x_i}A_j(x)-\partial_{x_j}A_i(x), \qquad i\neq j, \quad x\in \mathbb{R}^d. \] 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.