Exactly solvable Schrödinger eigenvalue problems for new anharmonic potentials with variable bumps and depths

Abstract

A new approach based on Darboux transformations is introduced to generate classes of solvable Schrödinger equations for new anharmonic potentials with variable bumps and depths. By introducing the concept of a transformation key, we present a method of controlling the number of bumps and their depths in these potentials. Although this method was applied to the one-dimensional generalized harmonic oscillator potential, it can be easily adapted to generate exactly solvable potentials using other known quantum potentials.

Appendix A: Continuation of transformations from Sect. 3

Here, we complete the transformations indicated by the key ${\mathcal {T}}_{m}=\{0;\,0,\,1,\,2,\,3,\,4\}$ for $m = 4,\,5,\,6$. Since the procedure is identical to the transformations for $m =2$, and $m=3$ as discussed in Sects. 2.2 and 2.3, only the final results shall be reported. In all cases, $E_{n} = a\left( 2n+1\right)$.

1.1 Fourth transformation ${\mathcal {T}}_{4}=2$

Seed Function: Square Integrable

$$\begin{aligned} \psi _{3;{\mathcal {T}}_{4}}(x)=-\frac{16 \left( 3 a^{3} \xi +2 a^{2} \xi ^{3}\right) }{a+2 \xi ^{2}} \exp \left[ -\frac{\xi ^{2}}{2 a}\right] \end{aligned}$$

(A.1)

Potential Energy: $x \in [-b/a,\infty )$

$$\begin{aligned} V_{4}(x)= \xi ^{2} + 4a + \frac{6 a^{2} \left( 3 a^{2}+4 \xi ^{4}\right) }{\xi ^{2} \left( 3 a+2 \xi ^{2}\right) ^{2}} \end{aligned}$$

(A.2)

Exact Solutions: $n=4,\,6,\,8,\,\ldots$

$$\begin{aligned} \psi _{4;n}(x)&= \exp \left[ \frac{\xi ^{2}}{2 a} \right] \left( D_{x}^{n+4}-\frac{3 \left( a^{2}-4 a \xi ^{2}-4 \xi ^{4}\right) }{\xi \left( 3 a+2 \xi ^{2}\right) } D_{x}^{n+3} \right. \nonumber \\&\qquad \qquad \qquad +\frac{6 \left( 3 a^{2}+4 a \xi ^{2}+4 \xi ^{4}\right) }{3 a+2 \xi ^{2}}D_{x}^{n+2}\nonumber \\&\qquad \qquad \qquad \left. -\frac{2 \left( 9 a^{3}-18 a^{2} \xi ^{2}-12 a \xi ^{4}-8 \xi ^{6}\right) }{\xi \left( 3 a+2 \xi ^{2}\right) } D_{x}^{n+1} \right) \nonumber \\&\quad \exp \left[ -\frac{\xi ^{2}}{a}\right] \end{aligned}$$

(A.3)

1.2 Fifth transformation ${\mathcal {T}}_{5}=3$

Seed Function: Nonsquare Integrable

$$\begin{aligned} \psi _{4;{\mathcal {T}}_{5}}(x)= \frac{48 \left( 3 a^{5}+12 a^{4} \xi ^{2}+4 a^{3} \xi ^{4}\right) }{\xi \left( 3 a+2 \xi ^{2}\right) } \exp \left[ -\frac{\xi ^{2}}{2 a}\right] \end{aligned}$$

(A.4)

Potential Energy: $x \in (-\infty ,\infty )$

$$\begin{aligned} V_{5}(x)= \xi ^{2} + 6a + \frac{16 a^{2} \left( -9 a^{3}+18 a^{2} \xi ^{2}+12 a \xi ^{4}+8 \xi ^{6}\right) }{\left( 3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}\right) ^{2}} \end{aligned}$$

(A.5)

Exact Solutions: $n=4,\,5,\,6,\,\ldots$

$$\begin{aligned} \psi _{5;n}&= \exp \left[ \frac{\xi ^{2}}{2 a}\right] \left( D_{x}^{n+5} + \frac{16 \left( 5 a \xi ^{3}+2 \xi ^{5}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+4} \right. \nonumber \\&\qquad \qquad \qquad + \frac{12 \left( 5 a^{3}+10 a^{2} \xi ^{2}+20 a \xi ^{4}+8 \xi ^{6}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+3} \nonumber \\&\qquad \qquad \qquad +\frac{32 \left( 15 a^{2} \xi ^{3}+12 a \xi ^{5}+4 \xi ^{7}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+2} \nonumber \\&\qquad \qquad \qquad \left. +\frac{4 \left( 45 a^{4}+120 a^{2} \xi ^{4}+64 a \xi ^{6}+16 \xi ^{8}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+1} \right) \nonumber \\&\quad \exp \left[ -\frac{\xi ^{2}}{a}\right] \end{aligned}$$

(A.6)

1.3 Sixth transformation ${\mathcal {T}}_{6}=4$

Seed Function: Square Integrable

$$\begin{aligned} \psi _{5;{\mathcal {T}}_{6}}(x)=-\frac{768 \left( 15 a^{6} \xi +20 a^{5} \xi ^{3}+4 a^{4} \xi ^{5}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} \exp \left[ -\frac{\xi ^{2}}{2 a}\right] \end{aligned}$$

(A.7)

Potential Energy: $x \in [-b/a,\infty )$

$$\begin{aligned} V_{6}(x)= \xi ^{2} + 8a + \frac{10 a^{2} \left( 45 a^{4}+120 a^{2} \xi ^{4}+64 a \xi ^{6}+16 \xi ^{8}\right) }{\xi ^{2} \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) ^{2}} \end{aligned}$$

(A.8)

Exact Solutions: $n=6,\,8,\,10,\,\ldots$

$$\begin{aligned} \psi _{6;n}&= \exp \left[ \frac{\xi ^{2}}{2 a}\right] \nonumber \\&\quad \left( D_{x}^{n+6} -\frac{5 \left( 3 a^{3}-18 a^{2} \xi ^{2}-36 a \xi ^{4}-8 \xi ^{6}\right) }{\xi \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) } D_{x}^{n+5} \right. \nonumber \\&\qquad +\frac{20 \left( 15 a^{3}+30 a^{2} \xi ^{2}+36 a \xi ^{4}+8 \xi ^{6}\right) }{15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+4} \nonumber \\&\qquad -\frac{20 \left( 15 a^{4}-60 a^{3} \xi ^{2}-120 a^{2} \xi ^{4}-80 a \xi ^{6}-16 \xi ^{8}\right) }{\xi \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) } D_{x}^{n+3} \nonumber \\&\qquad +\frac{20 \left( 45 a^{4}+120 a^{3} \xi ^{2}+216 a^{2} \xi ^{4}+96 a \xi ^{6}+16 \xi ^{8}\right) }{15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+2} \nonumber \\&\qquad \left. -\frac{4 \left( 225 a^{5}-450 a^{4} \xi ^{2}-600 a^{3} \xi ^{4}-720 a^{2} \xi ^{6}-240 a \xi ^{8}-32 \xi ^{10}\right) }{\xi \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) } D_{x}^{n+1} \right) \nonumber \\&\quad \exp \left[ -\frac{\xi ^{2}}{a}\right] \end{aligned}$$

(A.9)