Central force and the higher rank numerical range
Let \(r=r(\theta)\) be the orbit of a point mass under a central force \(f(r)=-1/r^3\) with angular momentum \(M\). Let \(p=M/\sqrt{M^2-1}\). It is shown that the orbit is a transcendental curve if \(p\) is irrational, and that the orbit is an algebraic curve \(F_A(1,x,y)=0\) for some \(2m \times 2m\) nilpotent Toeplitz matrix \(A\) if \(p=m/j_0\) is rational, where \[ F(t,x,y) = F_A(t,x,y)=\det(t I_n + x (A+A^{\ast})/2+y (A-A^{\ast})/(2 i)). \] Furthermore, the rank-\(k\) numerical range \(\Lambda_k(A)\) of this nilpotent Toeplitz matrix is examined, showing that he sum of numbers of flat portions on the boundary of \(\Lambda_k(A)\), \(k=1,\cdots,m\) is equal to \((m-1 (2 m-3))\).
scientific article; zbMATH DE number 6009679