Regularization of the singular inverse square potential in quantum mechanics with a minimal length

We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the $s$-wave bound states equation in terms of Heun’s functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.

Figure 1

$h\equiv F(a,b,c;\frac{2\omega -1}{2\omega })$ as a function of $\omega$, for $\kappa =3∕4$. All quantities $a,b,c,\omega ,\kappa$ are dimensionless.

Figure 2

$h\equiv F(a,b,c;\frac{2\omega -1}{2\omega })$ as a function of $\omega$, for $\kappa =2$. All quantities $a,b,c,\omega ,\kappa$ are dimensionless.

Figure 3

$h\equiv F(a,b,c;\frac{2\omega -1}{2\omega })$ as a function of $\omega$, for $\kappa =1∕20$. All quantities $a,b,c,\omega ,\kappa$ are dimensionless.