Bari-Markus property for Riesz projections of 1D periodic Dirac operators

The Dirac operators Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi], with $L^2$-potentials v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,\pi]), considered on $[0,\pi]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$S_N = \frac{1}{2\pi i} \int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \quad P_n = \frac{1}{2\pi i} \int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz$$ are well--defined for $|n| \geq N$ if $N$ is sufficiently large. It is proved that $$\sum_{|n| > N} \|P_n - P_n^0\|^2 < \infty,$$ where $P_n^0, n \in \mathbb{Z},$ are the Riesz projections of the free operator. Then, by the Bari--Markus criterion, the spectral Riesz decompositions $$f = S_N f + \sum_{|n| >N} P_n f, \quad \forall f \in L^2;$$ converge unconditionally in $L^2.

Bari-Markus property for Riesz projections of 1D periodic Dirac operators

The Dirac operators Ly = i ((1)(0) (0)(-1))dy/dx + v(x)y, y = ((y1)(y2)), x is an element of[0, pi], with L-2-potentials v(x) = ((0)(Q(x)) (P(x))(0)), P, Q is an element of L-2([0, pi]), considered on [0, pi] with periodic, antiperiodic or Dinchlet boundary conditions (bc), have discrete spectra, and the Riesz projections, S-N = 1/2 pi iota integral(vertical bar z vertical bar=N - 1/2) (z - L-bc)(-1) dz. p(n) = 1/2 pi iota integral(vertical bar z-n vertical bar=1/2) (z - L-bc)(-1) dz are well-defined for vertical bar n vertical bar >= N if N is sufficiently large. It is proved that Sigma(vertical bar n vertical bar>N) parallel to P-n - P-n(0)parallel to(2) < infinity, where P-n(0), n is an element of Z, are the Riesz projections of the free operator. Then, by the Ban Markus criterion, the spectral Riesz decompositions f = SN + Sigma(vertical bar n vertical bar>N) P(n)f, for all f is an element of L-2 converge unconditionally in L-2. (C) 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinho

Improved asymptotics of the spectral gap for the Mathieu operator

The Mathieu operator {equation*} L(y)=-y"+2a \cos{(2x)}y, \quad a\in \mathbb{C},\;a\neq 0, {equation*} considered with periodic or anti-periodic boundary conditions has, close to $n^2$ for large enough $n$, two periodic (if $n$ is even) or anti-periodic (if $n$ is odd) eigenvalues $\lambda_n^-$, $\lambda_n^+$. For fixed $a$, we show that {equation*} \lambda_n^+ - \lambda_n^-= \pm \frac{8(a/4)^n}{[(n-1)!]^2} [1 - \frac{a^2}{4n^3}+ O (\frac{1}{n^4})], \quad n\rightarrow\infty. {equation*} This result extends the asymptotic formula of Harrell-Avron-Simon, by providing more asymptotic terms