Authors: Spiros Konstantogiannis
The quasi-exact solvability of symmetrized quartic anharmonic oscillators has been studied first by Znojil  and then by Quesne . In this work, we examine the solvability of these models using, as basic parameter, the energy-dependent, constant (i.e. position-independent) term of a quotient polynomial. We examine the cases n=0 and n=1, and we show that our results are in agreement with those of Quesne. For n=2, following a different path from that of Znojil, we derive the cubic equation that our parameter satisfies and for the case it has a root at zero, we follow the zero root to obtain an even-parity, ground-state wave function and an odd-parity, third-excited-state wave function. As in the case of the sextic anharmonic oscillator , the straightforwardness and transparency of the analysis demonstrates the eligibility of the quotient polynomial as a solvability tool of polynomial oscillators.
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