A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality

This paper gives a second way to solve the one-dimensional minimization problem of the form :

$$\begin{aligned} \min _{f\not \equiv 0}\frac{\displaystyle \int \limits _0^\infty \left( f''\right) ^2x^{\mu +1}dx\int \limits _0^\infty \left( {x}^2\left( f'\right) ^2 -\varepsilon f^2\right) {{x}}^{\mu -1}d{x}}{\displaystyle \left( \int \limits _0^\infty \left( f'\right) ^2 {{x}}^{\mu }d{x}\right) ^2} \end{aligned}$$

for scalar-valued functions f on the half line, where \(f'\) and \(f''\) are its derivatives and \(\varepsilon \) and \(\mu \) are positive parameters with \(\varepsilon < \frac{\mu ^2}{4}.\) This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).