The uniform over the whole line estimates of spectral expansions related to the selfadjoint extensions of the Hill operator and of the Schrödinger operator with a bounded and measurable potential

We consider some properties of the spectral expansions related to selfadjoint extensions of the operator Hu = −u″ + q(x)u over the whole line in the case when q(x) is a continuous periodic function (the Hill operator) and in the case when q(x) is a bounded measurable function. This paper gives a brief description of results obtained in the following directions: the uniform over estimates of the generalized eigenfunctions, the uniform over estimates of the spectral function, the uniform over equiconvergence with the Fourier integral expansion, and the uniform over rate of convergence for functions from the Sobolev-Liouville classes.