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Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

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Abstract

We consider the Hill operator

$$Ly = - y^{\prime \prime} + v(x)y, \quad0 \leq x \leq \pi,$$

subject to periodic or antiperiodic boundary conditions, with potentials v which are trigonometric polynomials with nonzero coefficients, of the form

  1. (i)

    ae −2ix + be 2ix;

  2. (ii)

    ae −2ix + Be 4ix;

  3. (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 \({L^2 ([0,\pi], \mathbb{C})}\) 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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Authors and Affiliations

  1. Sabanci University, Orhanli, 34956, Tuzla, Istanbul, Turkey

    Plamen Djakov

  2. Department of Mathematics, The Ohio State University, 231 West 18th Ave, Columbus, OH, 43210, USA

    Boris Mityagin

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  1. Plamen Djakov
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  2. Boris Mityagin
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Djakov, P., Mityagin, B. Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials. Math. Ann. 351, 509–540 (2011). https://doi.org/10.1007/s00208-010-0612-5

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