Journal of Advances in Mathematics and Computer Science

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 the solution of differential equations. In this paper we derive the explicit form of one parameter family of orthonormal bases on space \(\mathit{L}^2\)(\(\mathbb{R}\)\(^+\)). The bases are formed by eigenvectors of the self-adjoint extension \(\mathit{H}_\xi\), parametrized by \(\xi\) \(\in\) \(\small\langle\)0,\(\pi\)), of differential expression \(\mathit{H} = -{ \mathit{d}^2 \over \mathit{d}^{x2}} +{ \mathit{x}^2 \over 4} \) together with the spectrum \(\sigma\)(\(\mathit{H}_\xi\)) on the space \(\mathit{L}^2\)(\(\mathbb{R}^+\)). For each \(\xi\) the set of eigenvectors form an orthonormal basis of \(\mathit{L}^2\)(\(\mathbb{R}\)\(^+\)). From the physical point of view, it is a solution of the Schrodinger equation of a harmonic oscillator on a semi-straight line. To correlate platelet count, splenic index (SI), platelet count/spleen diameter ratio and portal-systemic venous collaterals with the presence of esophageal varices in advanced liver disease to validate other screening parameters.