Orthonormal Bases onL2(R+)

In addition to orthogonal polynomials, orthogonal functions also play an important role. Their applications are, among others, in the fields of signal and data analysis, dynamic modeling. They are related to thesolution of differential equations. In this paper we derive the explicit form of one parameter family oforthonormal bases on space L2(R+).The bases are formed by eigenvectors of the self-adjoint extension Hξ, parametrized byξ∈ 〈0,π),of differential expression H=−d2dx2+x24 together with the spectrum σ(Hξ) on the space L2(R+).For each ξ the set of eigenvectors form an orthonormal basis of L2(R+). From thephysical point of view, it is a solution of the Schr ̈odinger equation of a harmonic oscillator on a semi-straightline. To correlate platelet count, splenic index (SI), platelet count/spleen diameter ratio and portal-systemicvenous collaterals with the presence of esophageal varices in advanced liver disease to validate other screeningparameters.