Divergence of spectral decompositions of Hill operators with two exponential term potentials
Djakov, Plamen Borissov and Mityagin, Boris Samuel (2013) Divergence of spectral decompositions of Hill operators with two exponential term potentials. Journal of Functional Analysis, 265 (4). pp. 660-685. ISSN 0022-1236
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Official URL: http://dx.doi.org/10.1016/j.jfa.2013.05.003
We consider the Hill operator Ly = -y '' + v(x)y, 0 <= x <= pi, subject to periodic or antiperiodic boundary conditions (bc) with potentials of the form v(x) = ae(-2irx) + be(2isx), a, b not equal 0, r, s is an element of N, r not equal s. It is shown that the system of root functions does not contain a basis in L-2([0, pi], C) if bc are periodic or if bc are antiperiodic and r, s are odd or r = 1 and s >= 3.
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