Riesz basis property of Hill operators with potentials in weighted spaces

P. Djakov, B. Mityagin
2014 Transactions of the Moscow Mathematical Society  
Consider the Hill operator L(v) = −d 2 /dx 2 +v(x) on [0, π] with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough n close to n 2 there are one Dirichlet eigenvalue μ n and two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λ − n , λ + n (counted with multiplicity). We describe classes of complex potentials v(x) = 2Z V (k)e ikx in weighted spaces (defined in terms of the Fourier coefficients of v) such that the periodic (or antiperiodic) root
more » ... iperiodic) root function system of L(v) contains a Riesz basis if and only if The theory of self-adjoint ordinary differential operators (o.d.o.) is well-developed, and the spectral decompositions play a central role in it [24, 29, 26] . Convergence of the spectral decompositions of non-self-adjoint o.d.o., considered on a finite interval I and subject to strictly regular boundary conditions (see [29, § 4.8]), has been understood completely in the early 1960's [27, 23, 17] . In this case, we not only have convergence, but the system of eigenfunctions (SEF) is a Riesz basis in L 2 (I). However, in the case of regular but not strictly regular boundary conditions-even in the case of periodic or antiperiodic boundary conditions-complete understanding appeared only in the 2000s as a result of the interaction of two lines of research. One stems from a question raised by A. Shkalikov in 1996A. Shkalikov in /1997 in the Kostyuchenko-Shkalikov seminar on Spectral Analysis at Moscow State University. Shkalikov formulated the following assertion and sketched an approach to its proof. Consider the Hill operator with a smooth potential q such that for some s ≥ 0 q (k) (0) = q (k) (π), 0 ≤ k ≤ s − 1, 2010 Mathematics Subject Classification. Primary 47E05, 34L40, 34L10.
doi:10.1090/s0077-1554-2014-00230-2 fatcat:7tg6v4zrzzftdbsy3ji4dkwo3m