Remote Access Transactions of the Moscow Mathematical Society

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
DOI: https://doi.org/10.1090/S0077-1554-2014-00230-2
Published electronically: November 4, 2014
MathSciNet review: 3308607
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

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


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 47E05, 34L40, 34L10

Retrieve articles in all journals with MSC (2010): 47E05, 34L40, 34L10


Additional Information

P. Djakov
Affiliation: Sabanci University, Orhanli, Istanbul, Turkey
Email: djakov@sabanciuniv.edu

B. Mityagin
Affiliation: Department of Mathematics, The Ohio State University
Email: mityagin.1@osu.edu

DOI: https://doi.org/10.1090/S0077-1554-2014-00230-2
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