7 research outputs found
Convergence Radii for Eigenvalues of Tri--diagonal Matrices
Consider a family of infinite tri--diagonal matrices of the form
where the matrix is diagonal with entries and the matrix
is off--diagonal, with nonzero entries The spectrum of is discrete. For small the
-th eigenvalue is a well--defined analytic
function. Let be the convergence radius of its Taylor's series about It is proved that R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq
\alpha <11/6.$
Equiconvergence of spectral decompositions of Hill operators
We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = -d (2)/dx (2) + v(x), x a L (1)([0, pi], with H (per) (-1) -potential and the free operator L (0) = -d (2)/dx (2), subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that
parallel to S-N - S-N(0) : L-a -> L-b parallel to -> 0 if 1 < a <= b < infinity, 1/a - 1/b < 1/2,
where S (N) and S (N) (0) are the N-th partial sums of the spectral decompositions of L and L (0). Moreover, if v a H (-alpha) with 1/2 < alpha < 1 and , then we obtain the uniform equiconvergence aEuro-S (N) -S (N) (0) : L (a) -> L (a)aEuro- -> 0 as N -> a