Generalized Bargmann inequalities (Q1057466)

The paper is concerned with the derivation of a set of inequalities which give lower bounds to \(\| \nabla \psi \|^ 2\) on \(L^ 2({\mathbb{R}}^ d)\). Their right-hand sides contain a real-valued function g; in the particular case \(g(x)=r^{\mu +1},\) we get the set of inequalities derived some years ago by \textit{V. Bargmann} [Helv. Phys. Acta 45, 249 (1972)]. This set covers both the uncertainty relations and the so-called local uncertainty principle. The most remarkable feature of the inequalities under consideration is that they give different estimates on the eigenspaces of the (generalized) angular momentum. The paper is concluded by a few simple examples. It is shown that a rigorous version of the inequality derived by Sachrajda, Weldon and Blanckenbecler is included in our set but their ''quasiclassical method'' of using it is in error. An example is also given to show a situation when a non-polynomial g gives better lower bounds to the eigenvalues than a polynomial one.