Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

This is the latest version of this item.

Abstract

We consider the Hill operator Ly=−y+v(x)y0x subject to periodic or antiperiodic boundary conditions, with potentials v which are trigonometric polynomials with nonzero coefficients, of the form (i) ae −2ix + be 2ix ; (ii) ae −2ix + Be 4ix ; (iii) ae −2ix + Ae −4ix + be 2ix + Be 4ix . Then the system of eigenfunctions and (at most finitely many) associated functions is complete but it is not a basis in L2([0]C) if |a| ≠ |b| in the case (i), if |A| ≠ |B| and neither −b 2/4B nor −a 2/4A is an integer square in the case (iii), and it is never a basis in the case (ii) subject to periodic boundary conditions.

Available Versions of this Item

• Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials. (deposited 08 Nov 2010 11:55)
  • Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials. (deposited 02 Dec 2011 11:41) [Currently Displayed]