Quasiexact solutions to a Schrödinger equation pertain to limited regions of a spectrum of eigenstates for which closedform eigenfunctions and eigenvalues can be derived, whereas the remainder of the spectrum can only be approximated. These occur for certain potentials with parameters in some limited range. The sextic anharmonic oscillator is the only onedimensional polynomial potential that can be quasiexactly solved if its parameters are appropriately chosen. Depending on the parameters, the system can be a single, double or triplewell potential. Consider solutions of the timeindependent onedimensional Schrödinger equation, in atomic units: . One trick for finding quasiexact solutions is to assume some appropriately behaved function and to use the relation to identify a potential function . For example, the simplest case of a quasiexact sextic anharmonic oscillator follows from , , . More generally, it can be shown that the potential can have the form , . For the cases , and , we show plots of the quasiexact eigenfunctions and of the potential functions , with superposed eigenvalues shown in red.
For simplicity, set . For , . The ground state is quasiexactly soluble: , , . For , . The firstexcited state is quasiexactly soluble: , , . For , . The ground state and the secondexcited state are quasiexactly soluble: , , , , .
