arXiv subjects

Evgeny Korotyaev

Publications and source records attributed to Evgeny Korotyaev.

100 records · Page 6Linked to original sources

arXiv2006

A priori estimates for conformal mappings on complex plane with parallel slits

We study the properties of a conformal mapping $z(k)$ from the plane without vertical slits $\G_n=[u_n-ih_n, u_n+ih_n], n\in\Z$ and $h=(h_n)_{n\in\Z}\in \ell^2$, onto the complex plane without horizontal slits $\g_nß\R, n\in\Z$, with the asymptotics $z(iv)=iv+ o(1), v\to\iy$. Here $u_{n+1}-u_n\ge 1, n\in \Z$. Introduce the sequences $l=(|\g_n|)_{n\in\Z}$. % where $J_n\ge 0,J_n^2=\int_{\G_n}|\Im z(k,h)||dk|/π$. We obtain a priori two-sided estimates for $\|h\|_{p,ø}, \|l\|_{p,ø}$, where %$\|h\|_ø^p$ is the norm of the Banach space %the extension of i)-ii) for the case $h\in\ell_ø^p$, where %$\ell_ø^p,1\le p\le 2$ with % the norm $\|h\|_{p,ø}^p=\sum ø_n|h_n|^p, 1\le p\le 2$ with any weight $ø_n\ge 1, n\in \Z$. Moreover, we determine other estimates.

math.CV↗

arXiv2006

The inverse problem for perturbed harmonic oscillator on the half-line with Dirichlet boundary conditions

We consider the perturbed harmonic oscillator $T_Dψ=-ψ''+x^2ψ+q(x)ψ$, $ψ(0)=0$, in $L^2(\R_+)$, where $q\in\bH_+=\{q', xq\in L^2(\R_+)\}$ is a real-valued potential. We prove that the mapping $q\mapsto{\rm spectral data}={\rm \{eigenvalues of\}T_D{\rm \}}\oplus{\rm \{norming constants\}}$ is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to $q\in\bH_+$ is given.

math.SP↗

arXiv2006

Schrödinger operators on zigzag graphs

We consider the Schrödinger operator on zigzag graphs with a periodic potential. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We obtain the following results from the inverse spectral theory: 1) we describe all finite gap potentials, 2) the mapping: potential -- all eigenvalues is a real analytic isomorphism for some class of potentials. We apply all these results to quasi-1D models of zigzag single-well carbon nanotubes.

math.SP↗

arXiv2006

Zigzag periodic nanotube in magnetic field

We consider the magnetic Schrödinger operator on the so-called zigzag periodic metric graph (a quasi 1D continuous model of zigzag nanotubes) with a periodic potential. The magnetic field (with the amplitude $B\in R$) is uniform and it is parallel to the axis of the nanotube. The spectrum of this operator consists of an absolutely continuous part (spectral bands separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that endpoints of the gaps are periodic or anti-periodic eigenvalues or resonances (real branch points of the Lyapunov function). We describe the spectrum as functions of $B$. For example, if $B\to B_{k,m}=\frac{4(\fracπ{2}-\frac{πk}{N}+πm)}{\sqrt 3 \cos \fracπ{2N}}, k=1,2,..,N, m\in Z$, then some spectral band shrinkes into a flat band i.e., an eigenvalue of infinite multiplicity.

math.SP↗

arXiv2005

The inverse problem for perturbed harmonic oscillator on the half-line

We consider the perturbed harmonic oscillator $T_Dψ=-ψ''+x^2ψ+q(x)ψ$, $ψ(0)=0$ in $L^2(R_+)$, where $q\in H_+=\{q', xq\in L^2(R_+)\}$ is a real-valued potential. We prove that the mapping $q\mapsto{\rm spectral data}={\rm \{eigenvalues of\}T_D{\rm \}}\oplus{\rm \{norming constants\}}$ is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to $q\in H_+$ is given. Moreover, we solve the similar inverse problem for the family of boundary conditions $ψ'(0)=b ψ(0)$, $b\in R$.

math-ph↗

arXiv2005

The Lyapunov function for Schrödinger operators with a periodic 2x2 matrix potential

We consider the Schrödinger operator on the real line with a 2x2 matrix valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: 1) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.

math.SP↗

arXiv2005

Spectral estimates for Schrödinger operator with periodic matrix potentials on the real line

We consider the Schrödinger operator on the real line with a $N\ts N$ matrix valued periodic potential, N>1. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy (in terms of the Fourier coefficients of the potential). We show that there exist two types of gaps: i) stable gaps, i.e., the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). Moreover, the following results are obtained: 1) we define the quasimomentum as an analytic function on the Riemann surface of the Lyapunov function; various properties and estimates of the quasimomentum are obtained, 2) we construct the conformal mapping with real part given by the integrated density of states and imaginary part given by the Lyapunov exponent. We obtain various properties of this conformal mapping, which are similar to the case N=1, 3) we determine various new trace formulae for potentials, the integrated density of states and the Lyapunov exponent, 4) a priori estimates of gap lengths in terms of potentials are obtained.

math.SP↗

arXiv2005

Schrödinger operator with a junction of two 1-dimensional periodic potentials

The spectral properties of the Schrödinger operator $T_ty= -y''+q_ty$ in $L^2(\R)$ are studied, with a potential $q_t(x)=p_1(x), x<0, $ and $q_t(x)=p(x+t), x>0, $ where $p_1, p$ are periodic potentials and $t\in \R$ is a parameter of dislocation. Under some conditions there exist simultaneously gaps in the continuous spectrum of $T_0$ and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of $T_t$. The following results are obtained: i) In any gap of $T_t$ there exist $0,1$ or 2 eigenvalues. Potentials with 0,1 or 2 eigenvalues in the gap are constructed. ii) The dislocation, i.e. the case $p_1=p$ is studied. If $t\to 0$, then in any gap in the spectrum there exist both eigenvalues ($ \le 2 $) and resonances ($ \le 2 $) of $T_t$ which belong to a gap on the second sheet and their asymptotics as $t\to 0 $ are determined. iii) The eigenvalues of the half-solid, i.e. $p_1={\rm constant}$, are also studied. iv) We prove that for any even 1-periodic potential $p$ and any sequences $\{d_n\}_1^{\iy}$, where $d_n=1$ or $d_n=0$ there exists a unique even 1-periodic potential $p_1$ with the same gaps and $d_n$ eigenvalues of $T_0$ in the n-th gap for each $n\ge 1.$

math.SP↗

arXiv2005

Spectral asymptotics for periodic fourth order operators

We consider the operator ${d^4dt^4}+V$ on the real line with a real periodic potential $V$. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: 1) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function above the bottom of the spectrum). We also show that the periodic and anti-periodic spectrum together determine the spectrum of our operator. Finally, we show that for small potentials $V\ne 0$ the spectrum in the lowest band has multiplicity 4 and the bottom of the spectrum is a resonance, and not a periodic (or anti-periodic) eigenvalue.

math.SP↗

arXiv2005

Inverse problem for the discrete 1D Schrödinger operator with small periodic potentials

Consider the discrete 1D Schrödinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of $2^k$ distinct potentials. Finally, the asymptotics of the spectrum are determined as $q\to 0$.

math.SP↗