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

The one-dimensional Schr ̈odinger's equation is analysed w ith regard to the existence of exact solutions for decatic polynomial potentials. Under c ertain conditions on the potential's pa- rameters, we show that the decatic polynomial potential V ( x ) = ax 10 + bx 8 + cx 6 + dx 4 + ex 2 , a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditi ons for corresponding energy-dependent polynomial solutions are given in detail. It is also shown th at these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy e igenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schr ̈odinger equation. Further analysis for arbitrary values of the potential para meters using the asymptotic iteration method is also presented