The Zero Set of a Real Analytic Function

A brief proof of the statement that the zero-set of a nontrivial real-analytic function in $d$-dimensional space has zero measure is provided.Comment: 4 page

Equiconvergence of spectral decompositions of Hill-Schr\"odinger operators

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator $L= -d^2/dx^2 + v(x),$ $x \in [0,\pi],$ with $H_{per}^{-1}$-potential and the free operator $L^0=-d^2/dx^2,$ subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $$\|S_N - S_N^0: L^a \to L^b \| \to 0 \quad \text{if} \;\; 1<a \leq b< \infty, \;\; 1/a - 1/b <1/2,$$ where $S_N$ and $S_N^0$ are the $N$-th partial sums of the spectral decompositions of $L$ and $L^0.$ Moreover, if $v \in H^{-\alpha}$ with $1/2 < \alpha < 1$ and $\frac{1}{a}=(3/2)-\alpha,$ then we obtain uniform equiconvergence: $\|S_N - S_N^0: L^a \to L^\infty \| \to 0$ as $N \to \infty.

Combinatorial identities related to eigenfunction decompositions of Hill operators: Open Questions

We formulate several open questions related to enumerative combinatorics, which arise in the spectral analysis of Hill operators with trigonometric polynomial potentials

Criterion for Zd\mathbb{Z}_d--symmetry of a Spectrum of a Compact Operator

If $A$ is a compact operator in a Banach space and some power $A^q$ is nuclear we give a criterion of $\mathbb{Z}_{d}$ -- symmetry of its spectrum $\sigma{A}$ in terms of vanishing of the traces $\mathop{\mathit{Trace}} A^n$ for all $n$, $n \geq 0$, $n \neq 0 \mod d$, sufficiently large.Comment: 11 pages, no figure

Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert space $H$ with discrete spectrum $\{t_k \}$, $T \phi_k = t_k \phi_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator operator. In particular, if $t_{k+1}-t_k \geq c k^{\alpha - 1}, \quad \alpha > 1/2$ and $\| B \phi_k \| = o(k^{\alpha - 1})$ then the system of root vectors of $T+B$, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in $H$

Fourier method for one dimensional Schr\"odinger operators with singular periodic potentials

By using quasi--derivatives, we develop a Fourier method for studying the spectral properties of one dimensional Schr\"odinger operators with periodic singular potentials

Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators

Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,\pi].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2$ in the Hill case, or close to $n, \; n\in \mathbb{Z}$ in the Dirac case, 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 give estimates for the asymptotics of the spectral gaps $\gamma_n = \lambda_n^+ - \lambda_n^-$ and deviations $\delta_n =\mu_n - \lambda_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, for special potentials that are trigonometric polynomials we provide precise asymptotics of $\gamma_n$ and $\delta_n.

Divergence of spectral decompositions of Hill operators with two exponential term potentials

We consider the Hill operator $$Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq \pi,$$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$v(x) = a e^{-2irx} + b e^{2isx}, \quad a, b \neq 0, r,s \in \mathbb{N}, r\neq s.$$ It is shown that the system of root functions does not contain a basis in $L^2 ([0,\pi], \mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \geq 3.

Asymptotics of instability zones of the Hill operator with a two term potential

Let $\gamma_n$ denote the length of the $n$-th zone of instability of the Hill operator $Ly= -y^{\prime \prime} - [4t\alpha \cos2x + 2 \alpha^2 \cos 4x ] y,$ where $\alpha \neq 0,$ and either both $\alpha, t$ are real, or both are pure imaginary numbers. For even $n$ we prove: if $t, n$ are fixed, then, for $\alpha \to 0,$ $$\gamma_n = | \frac{8\alpha^n}{2^n [(n-1)!]^2} \prod_{k=1}^{n/2} (t^2 - (2k-1)^2) | (1 + O(\alpha)),$$ and if $\alpha, t$ are fixed, then, for $n \to \infty,$ $$\gamma_n = \frac{8 |\alpha/2|^n}{[2 \cdot 4 ... (n-2)]^2} | \cos (\frac{\pi}{2} t) | [ 1 + O (\frac{\log n}{n}) ].$$ Similar formulae (see Theorems \ref{thm2} and \ref{thm4}) hold for odd $n.$ The asymptotics for $\alpha \to 0$ imply interesting identities for squares of integers.Comment: 39 page

Eigensystem of an L2L^2-perturbed harmonic oscillator is an unconditional basis

We prove the following. For any complex valued $L^p$-function $b(x)$, $2 \leq p < \infty$ or $L^\infty$-function with the norm $\| b | L^{\infty}\| < 1$, the spectrum of a perturbed harmonic oscillator operator $L = -d^2/dx^2 + x^2 + b(x)$ in $L^2(\mathbb{R}^1)$ is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in $L^2(\mathbb{R})$.Comment: 28 page