Physics – Mathematical Physics
Scientific paper
2011-04-03
Physics
Mathematical Physics
19 pages
Scientific paper
We study Hermitian unitary matrices $\S\in\C^{n,n}$ with the following property: There exist $r\geq0$ and $t>0$ such that the entries of $\S$ satisfy $|\S_{jj}|=r$ and $|\S_{jk}|=t$ for all $j,k=1,...,n$, $j\neq k$. We derive necessary conditions on the ratio $d:=r/t$ and show that they are very restrictive except for the case when $n$ is even and the sum of the diagonal elements of $\S$ is zero. Examples of families of matrices $\S$ are constructed for $d$ belonging to certain intervals. The case of real matrices $\S$ is examined in more detail. It is demonstrated that a real $\S$ can exist only for $d=n/2-1$, or for even $n$'s and $d$ satisfying $n/2+d\equiv1\, (\mod2)$. We provide a detailed description of the structure of real $\S$ with $d\geq n/4-3/2$, and derive a sufficient and necessary condition of their existence in terms of the existence of certain symmetric $(v,k,\lambda)$-designs. We prove that there exist no real $\S$ with $d\in(n/6-1,n/4-3/2})$. A parametrization of Hermitian unitary matrices is also proposed, and its generalization to general unitary matrices is given. At the end of the paper, the role of the studied matrices in quantum mechanics on graphs is briefly explained.
Cheon Taksu
Turek Ondrej
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