Divergence property of Fourier and Ritz expansions

A divergence property of approximating sequences ψₙ=∑ₘ₌₁ⁿcₙₘφₘ that converge in the norm of the Hilbert space L₂(RN) to a fast-decay function ψ is studied. The expansion ψₙ can be a Fourier one or obtained by solving an eigenproblem by the Ritz method and the basis set φₘ need not be orthogonal in L₂(RN). The notion of uniform boundedness is used to show that if ψₙ is nonuniformly bounded, then it diverges from its correct limit ψ in such a way that there is an increasing separation between the asymptotic tails of ψₙ and ψ as n increases. The analytical and numerical examples show that the rate of this divergence may be exponential, hence the divergence of expectation-value sequences S(ψₙ) is proved for some operators S whose correct expectation value S(ψ) depends mainly on the long-range behavior of ψ. The compatibility between several convergence properties of the approximating sequence ψₙ and basis set properties with the nonuniform boundedness property is shown. We show that the well-known property of some trial wave functions to generate correct expectation values of some operators and incorrect values for other operators is connected with the property of nonuniformly bounded sequences ψₙ to converge correctly on a finite region and diverge on its complementary one, hence it is proved that correct expectation values can be obtained from a nonuniformly bounded sequence by using a suitable limiting procedure. As model examples, Fourier and Ritz expansions of the ground state ψ=2Z3/2e^-Zr of a hydrogenlike atom are considered.