James Womack
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 Nov 24 awarded Popular Question Jul 2 awarded Curious Jan 7 comment Why is the double factorial $(-1)!! = 1$, by definition? +1 for extending the answer to cover the even number double factorials, too. Jan 7 accepted Why is the double factorial $(-1)!! = 1$, by definition? Jan 7 awarded Nice Answer Jan 7 awarded Yearling Jan 6 awarded Self-Learner Jan 6 awarded Commentator Jan 6 comment Why is the double factorial $(-1)!! = 1$, by definition? I couldn't find any specific answers to this on math.stackexchange.com, but worked out an answer for myself. Therefore, I've answered the question myself. Jan 6 asked Why is the double factorial $(-1)!! = 1$, by definition? Jan 6 answered Why is the double factorial $(-1)!! = 1$, by definition? Aug 12 comment What are the requirements for a “test” function to show operators commute? Without introducing a test function, I do not know how I would demonstrate commutativity. I say that I only know the action of $\hat{H}_{0}$ on determinants/products of the basis functions $\chi_{a}$ because I know that $\hat{f}_{i}\chi_{a}(\mathbf{x}_{i}) = \epsilon_{a} \chi_{a}(\mathbf{x}_{i})$ (an eigenvalue problem) but not the outcome of $\hat{f}_{i}$ acting on any arbitrary function of $\mathbf{x}_{i}$ (not necessarily an eigenfunction of $\hat{f}_{i}$). What do you mean by "commutativity for the entire space follows by linearity"? Aug 9 asked What are the requirements for a “test” function to show operators commute? Dec 7 comment Does the locality or non-locality of operators imply matrix structure? I can see then, how this question is not really meaningful. Thanks for your help. I'll leave this for now and maybe come back to it if I have a specific approximation I'd like to discuss. Dec 6 comment Does the locality or non-locality of operators imply matrix structure? @joriki So, you are saying that I would need to explicitly state the approximation used to discretize the continuous operator to make any statement about the structure of the matrix? Dec 6 asked Does the locality or non-locality of operators imply matrix structure? Nov 22 answered Matrix elements of an inverse Hermitian matrix Nov 22 revised Matrix elements of an inverse Hermitian matrix edited so the question is less specific and potentially more useful to others Nov 22 comment Matrix elements of an inverse Hermitian matrix Ah thanks. I knew this, but I think what I missed that not only is $(\omega\mathbf{1}-\mathbf{A})^{1}$ a function of $(\omega\mathbf{1}-\mathbf{A})$, it is also a function of $\mathbf{A}$. I'll work though the problem and write up the solution as an answer if I get there. Nov 21 asked Matrix elements of an inverse Hermitian matrix