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- W1976532654 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 $v(x)= sum_{2mathbb{Z}} V(k) e^{ikx}$ 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 $$V(-2n) asymp V(2n) quad text{as} ;; n in 2mathbb{N};; (text{or} ; n in 1+ 2mathbb{N}), ;; n to infty.$$ For such potentials we prove that $lambda_n^+ - lambda_n^- sim pm 2sqrt{V(-2n)V(2n)} $ and $$mu_n - frac{1}{2}(lambda_n^+ + lambda_n^-) sim -frac{1}{2} (V(-2n) + V(2n)).$$" @default.
- W1976532654 created "2016-06-24" @default.
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- W1976532654 date "2014-03-12" @default.
- W1976532654 modified "2023-09-27" @default.
- W1976532654 title "Riesz basis property of Hill operators with potentials in weighted spaces" @default.
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