On the behaviour of functions at the boundary conditions in the domain of the generalised momentum operators

Abstract

In this paper we have investigated the general condition of self-adjointness of the generalised momentum operators and we have shown that it highly depends on the metric of the space. We have also discussed the domain of the generalised momentum operators at boundary conditions.

Appendix A

The Cauchy–Schwarz inequality is a result of the inner product being positive-definite which is part of the definition of the inner product. At first, consider the scalar product in curved space

$$\begin{aligned} (f,h)=\int _a^b\sqrt{(}g)f^*(x)h(x)\mathrm {d}^nx. \end{aligned}$$

(A.1)

Suppose

$$\begin{aligned} A=(f,f) , B=(h,h), C=(h,f). \end{aligned}$$

(A.2)

The inequality is $AB\ge |C|^2$. For

$$\begin{aligned} B=0\Leftrightarrow h=0\Rightarrow C=0. \end{aligned}$$

(A.3)

The inequality is satisfied. Therefore, we have to prove only for the case when $B\ne 0$.

$$\begin{aligned} \int _a^b\sqrt{(}g)|Bf(x)-Ch(x)|^2=B(BA-|C|^2)\ge 0.\nonumber \\ \end{aligned}$$

(A.4)

As $B\ne 0$ and $B>0$ we have $BA\ge |C|^2$.