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

Lower bound for solutions to the secular equation in a direct sum of the Sobolev spaces of divided regions