  In this paper a lower bound for solutions to the secular equation of the
Schr\"odinger equation with basis functions discontinuous on boundaries of
divided regions is given. If the functions do not have the discontinuity, the
bound reduces to that for the usual Rayleigh-Ritz method. Difference from the
usual Rayleigh-Ritz method is bounded by the degree of discontinuity. The
result can be regarded as a theoretical basis of the augmented plane wave
method for the band structure calculations in solid state physics. The result
would be useful also for other electronic eigenvalue problems in which the
behavior of the eigenfunction near nuclei is very different from that in the
interstitial region, because it allows us to use different basis functions in
different regions in contrast with the usual Rayleigh-Ritz method in which we
need to use basis functions with inappropriate behaviors in wide regions. The
proof is based only on new general results about the Sobolev space, in
particular, equivalence of two semi-norms concerned with discontinuity of the
functions in a direct sum of the Sobolev spaces. Therefore, the result does not
depend on specific forms of basis functions at all. The main ingredient of the
proof is an estimate of elements of the orthogonal complement of
$H^1_0(\Omega)$ by their boundary values based on a characterization of them as
weak solutions to an elliptic equation on a region $\Omega$.
