What is the domain of the self-adjoint ( Hermitian) operator of $q(x)=xf(x)$. in $L^2[a,b]$, $(-\infty < a < b < \infty)$. What if $a= -\infty$ and/or $b=\infty$ what would be the new domain? I have already demonstrated that the operator is self-adjoint and I want to know what is $\int x^2f^2(x)dx$ from a to be to know the bounds. What if the bounds were from negative infinity to positive infinity. Is there an equality that I have to use?
Self-adjoint (Hermitian) Position Operator
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linear-algebra
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0The new domain would evaluate to zero. – 2017-02-14
1 Answers
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If $a$ and $b$ are finite the operator "$x$" is bounded,and therefore definable on all of $L^2[a,b]$. If the interval is infinite you should restrict to $f(x)$ such that $xf(x)\in L^2$. I doubt that there is a more specific way of expressing this condition other than the one you give --- that $\int x^2|f|^2 dx$ be finite.