Quasi-Exact Solutions of Schrödinger Equation: Sextic Anharmonic Oscillator

Quasi-exact solutions to a Schrödinger equation pertain to limited regions of a spectrum of eigenstates for which closed-form 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 one-dimensional polynomial potential that can be quasi-exactly solved if its parameters are appropriately chosen. Depending on the parameters, the system can be a single-, double- or triple-well potential.
Consider solutions of the time-independent one-dimensional Schrödinger equation, in atomic units:
.
One trick for finding quasi-exact 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 quasi-exact sextic anharmonic oscillator follows from
,
which gives
,
with
.
More generally, it can be shown that the potential can have the form
, .
For the cases , and , we show plots of the quasi-exact eigenfunctions and of the potential functions , with superposed eigenvalues shown in red.

SNAPSHOTS

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DETAILS

For simplicity, set . For ,
.
The ground state is quasi-exactly soluble:
, ,
.
For ,
.
The first-excited state is quasi-exactly soluble:
, ,
.
For ,
.
The ground state and the second-excited state are quasi-exactly soluble:
, ,
, ,
.
References
[1] A. V. Turbiner, "One-Dimensional Quasi-exactly Solvable Schrödinger Equations," Physics Reports, 642, 2016 pp. 1–71. doi:10.1016/j.physrep.2016.06.002.
[2] S. Konstantogiannis, "Quasi-exact Solution of Sextic Anharmonic Oscillator Using a Quotient Polynomial," Journal for Foundations and Applications of Physics, 6(1), 2019 pp. 17–34. sciencefront.org/ojs/index.php/jfap/article/view/109.
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