On a central algorithm for calculation of the inverse of the harmonic oscillator in the spaces of orbits

Abstract

The equation Au=f with a linear symmetric positive definite operator A:D(A)⊂H→H having a discrete spectrum and dense image in a complex Hilbert space H is considered. This equation is transferred into the Hilbert space of finite orbits D(An) as well as into the Fréchet space of all orbits D(A∞), that is, the projective limit of the sequence of spaces {D(An)}. For an approximate solution of the inverse of A, linear spline central algorithms in these spaces are constructed. The convergence of the sequence of approximate solutions to the exact solution is proved. The obtained results are applied to the quantum harmonic oscillator operator Au(t)=−u″(t)+t2u(t), t∈R, in the Hilbert space of finite orbits D(An), and in the Fréchet space of all orbits D(A∞) that in this case coincides with the Schwartz space of rapidly decreasing functions. Some quantum mechanical interpretations of obtained results are also given.