Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential

Table 1

Conditions on the value of the parameters 𝑔 and 𝜇 for the quasipolynomial solutions in the case of Δ₂=0 with different values of 𝑤𝑎² and 𝑙.

𝑛	𝑙	𝑤𝑎^2	Conditions	𝐸^𝑤𝑎2𝑛,𝑙≡𝐸^𝑤𝑎2𝑛,𝑙(𝜇,𝑔)
1	−1	1/2	𝜇=1/3(−3−15𝐴^−1/3−𝐴^1/3),𝐴=3(36−√961)	𝐸^1/21,−1=−𝑤(3/2+2/3𝐴^1/3+10𝐴^−1/3)
			𝑔=1/9𝐴^−2/3(15+6𝐴^1/3+𝐴^2/3)(15+9𝐴^1/3+𝐴^2/3)
		1	𝜇=1/3(−5−19𝐴^−1/3−𝐴^1/3),𝐴=161−3√2118	𝐸^11,−1=−𝑤(17/6+2/3𝐴^1/3+38/3𝐴^−1/3)
			𝑔=1/9𝐴^−2/3(19+8𝐴^1/3+𝐴^2/3)(19+11𝐴^1/3+𝐴^2/3)
		3/2	𝜇=1/3(−7−25𝐴^−1/3−𝐴^1/3),𝐴=199−18√74	𝐸^3/21,−1=−𝑤(25/6+2/3𝐴^1/3+50/3𝐴^−1/3)
			𝑔=1/9𝐴^−2/3(25+10𝐴^1/3+𝐴^2/3)(25+13𝐴^1/3+𝐴^2/3)
		2	𝜇=1/3(−9−33𝐴^−1/3−𝐴^1/3),𝐴=3(72−√1191)	𝐸^21,−1=−𝑤(11/2+2/3𝐴^1/3+22𝐴^−1/3)
			𝑔=1/9𝐴^−2/3(33+12𝐴^1/3+𝐴^2/3)(33+15𝐴^1/3+𝐴^2/3)
	0	1/2	𝜇=0	𝐸^1/21,0=3/2𝑤
			𝑔=2
			𝜇=−1/2(7+√17)	𝐸^1/21,0=−1/2(11+2√17)𝑤
			𝑔=29+5√17
			𝜇=−1/2(7−√17)	𝐸^1/21,0=−1/2(11−2√17)𝑤
			𝑔=29−5√17
		1	𝜇=−3+𝐵	𝐸^11,0=−(9/2−2𝐵)𝑤
			𝑔=(−4+𝐵)(−5+𝐵)
			𝐵=1/3ℜ(𝐴^1/3+33𝐴^−1/3),𝐴=−108+3𝑖√2697
			𝜇=−3−𝐵,	𝐸^11,0=−(9/2+2𝐵)𝑤
			𝑔=(5+𝐵)(4+𝐵)
			𝐵=ℜ((11(1+𝑖√3)𝐴^−1/3/2)+((1−𝑖√3)𝐴^1/3/6)),𝐴=−108+3𝑖√2697
			𝜇=−3−𝐵,	𝐸^11,0=−(9/2+2𝐵)𝑤
			𝑔=(5+𝐵)(4+𝐵)
			𝐵=ℜ(11(1−𝑖√3)𝐴^−1/3/2+((1+𝑖√3)𝐴^1/3/6)),𝐴=−108+3𝑖√2697