Invariant relation in the problem of size quantization in a self-consistent potential near the semiconductor boundary

The invariant relation in the eigenvalue problem is found for a one-dimensional nonlinear integrodifferential operator acting on the normalized eigenfunctions. The invariance of the obtained relationship follows, in particular, from the fact that it is independent of the detailed form of the operator. This problem arises in the physics of a two-dimensional electron system near the semiconductor boundary, where the potential well is formed, on the one hand, by a high potential barrier at the boundary and, on the other, by the intrinsic electric field screened by two-dimensional electrons filling the well. The invariant relates the energy of the size-quantization level to the average size of electron wave function in the well.