Exact and approximate solutions to Schroedinger's equation with decatic potentials

摘要

The one-dimensional Schroedinger's equation is analysed with regard to theexistence of exact solutions for decatic polynomial potentials. Under certainconditions on the potential's parameters, we show that the decatic polynomialpotential $V(x)=ax^{10}+bx^8+cx^6+dx^4+ex^2$, $a>0$ is exactly solvable. Byexamining the polynomial solutions of certain linear differential equationswith polynomial coefficients, the necessary and sufficient conditions forcorresponding energy-dependent polynomial solutions are given in detail. It isalso shown that these polynomials satisfy a four-term recurrence relation,whose real roots are the exact energy eigenvalues. Further, it is shown thatthese polynomials generate the eigenfunction solutions of the correspondingSchroedinger equation. Further analysis for arbitrary values of the potentialparameters using the asymptotic iteration method is also presented.