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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



Riesz basis property of Hill operators with potentials in weighted spaces

Authors: P. Djakov and B. Mityagin
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2014, 151-172
MSC (2010): Primary 47E05, 34L40, 34L10
Published electronically: November 4, 2014
MathSciNet review: 3308607
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Abstract: Consider the Hill operator $ L(v) = - d^2/dx^2 + v(x) $ on $ [0,\pi ]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $ n$ close to $ n^2 $ there are one Dirichlet eigenvalue $ \mu _n$ and two periodic (if $ n$ is even) or antiperiodic (if $ n$ is odd) eigenvalues $ \lambda _n^-, \lambda _n^+ $ (counted with multiplicity).

We describe classes of complex potentials $ v(x)= \sum \nolimits _{\kern 1pt 2\mathbb{Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $ v$) such that the periodic (or antiperiodic) root function system of $ L(v) $ contains a Riesz basis if and only if

$\displaystyle V(-2n) \asymp V(2n)$$\displaystyle \quad \text {as}~n \in 2\mathbb{N}~(\text {or}~n \in 1+ 2\mathbb{N}),\quad n \to \infty .$

For such potentials we prove that $ \lambda _n^+ - \lambda _n^- \sim \pm 2\sqrt {V(-2n)V(2n)}$ and

$\displaystyle \mu _n - \frac {1}{2}(\lambda _n^+ + \lambda _n^-) \sim -\frac {1}{2} (V(-2n) + V(2n)).$

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Additional Information

P. Djakov
Affiliation: Sabanci University, Orhanli, Istanbul, Turkey

B. Mityagin
Affiliation: Department of Mathematics, The Ohio State University

Keywords: Hill operator, periodic and antiperiodic boundary conditions, Riesz bases
Published electronically: November 4, 2014
Additional Notes: P. Djakov acknowledges the hospitality of the Department of Mathematics of the Ohio State University, July–August 2013.
B. Mityagin acknowledges the support of the Scientific and Technological Research Council of Turkey and the hospitality of Sabanci University, May–June, 2013.
Dedicated: Dedicated to the memory of Boris Moiseevich Levitan on the occasion of the 100th anniversary of his birthday
Article copyright: © Copyright 2014 P. Djakov and B. Mityagin

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