Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials
Djakov, Plamen Borissov and Mityagin, Boris (2011) Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials. Mathematische Annalen, 351 (3). pp. 509-540. ISSN 0025-5831 (print) ; 1432-1807 (online)
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Official URL: http://dx.doi.org/10.1007/s00208-010-0612-5
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(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.
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