Results 1  10
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27
The estimates of periodic potentials in terms of effective masses,
 Comm. Math. Phys.,
, 1997
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Spectral estimates for periodic Jacobi matrices
, 2008
"... We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ 2 (Z) of the form (Hψ)n = an−1ψn−1 + bnψn + anψn+1, where an = an+q and bn = bn+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding t ..."
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Cited by 9 (4 self)
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We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ 2 (Z) of the form (Hψ)n = an−1ψn−1 + bnψn + anψn+1, where an = an+q and bn = bn+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H.
Inverse Problems Generated By Conformal Mappings on Complex Plane With Parallel Slits
, 2000
"... We study the properties of a conformal mapping z(k; h) from K(h) = C n [\Gamma n where \Gamma n = [u n \Gamma ijh n j; u n + ijh n j]; n 2 Z is a vertical slit and h = fh n g 2 ` 2 R , onto the complex plane with horizontal slits fl n ae R; n 2 Z, with the asymptotics z(iv; h) = iv + (iQ 0 (h) + ..."
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Cited by 8 (3 self)
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We study the properties of a conformal mapping z(k; h) from K(h) = C n [\Gamma n where \Gamma n = [u n \Gamma ijh n j; u n + ijh n j]; n 2 Z is a vertical slit and h = fh n g 2 ` 2 R , onto the complex plane with horizontal slits fl n ae R; n 2 Z, with the asymptotics z(iv; h) = iv + (iQ 0 (h) + o(1))=v; v ! +1. Here u n+1 \Gamma u n ? 1; n 2 Z, and the Dirichlet integral Q 0 (h) = RR C jz 0 (k; h) \Gamma 1j 2 dudv=(2) ! 1; k = u + iv. Introduce the sequences l = fl n g; J = fJ n g; where l n = jfl n j sign h n , and J n = jJ n j sign h n ; J 2 n = R \Gamma n j Im z(k; h)jjdkj=. The following results are obtain: 1) an analytic continuation of the function z(\Delta; \Delta) : K(h) \Theta ff : kf \Gamma hk ! rg ! C onto the domain K(h) \Theta ff : kf \Gamma hkC ! rg for h 2 ` 2 R and some r ? 0, and the Lowner equation for z(k; h) when the height of some slit h n is changed, 2) an analytic continuation of the functional Q 0 : ` 2 R ! R+ in the domain ff : k Im fk ! r...
Inverse Problem and Estimates for Periodic ZakharovShabat Systems
, 2000
"... Consider the ZakharovShabat (or Dirac) operator T zs on L 2 (R) \Phi L 2 (R) with real periodic vector potential q = (q 1 ; q 2 ) 2 H = L 2 (T) \Phi L 2 (T). The spectrum of T zs is absolutely continuous and consists of intervals separated by gaps (z \Gamma n ; z + n ); n 2 Z. ?From ..."
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Cited by 7 (4 self)
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Consider the ZakharovShabat (or Dirac) operator T zs on L 2 (R) \Phi L 2 (R) with real periodic vector potential q = (q 1 ; q 2 ) 2 H = L 2 (T) \Phi L 2 (T). The spectrum of T zs is absolutely continuous and consists of intervals separated by gaps (z \Gamma n ; z + n ); n 2 Z. ?From the Dirichlet eigenvalues m n ; n 2 Z of the ZakharovShabat equation with Dirichlet boundary conditions at 0; 1, the center of the gap and the square of the gap length we construct the gap length mapping g : H ! ` 2 \Phi` 2 . Using nonlinear functional analysis in Hilbert spaces, we show that this mapping is a real analytic isomorphism. Our proof relies on new identities and estimates contained in the second part of the our paper.
Effective masses for zigzag nanotubes in magnetic fields
, 2007
"... We consider the Schrödinger operator with a periodic potential on quasi1D models of zigzag singlewall carbon nanotubes in magnetic field. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multi ..."
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Cited by 6 (6 self)
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We consider the Schrödinger operator with a periodic potential on quasi1D models of zigzag singlewall carbon nanotubes in magnetic field. 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 obtain identities and a priori estimates in terms of effective masses and gap lengths. 1 Introduction and main results We consider the Schrödinger operator HB = (−i ∇ − A) 2 + Vq with a periodic potential Vq on the zigzag nanotube ΓN ⊂ R3 (1D models of zigzag singlewell carbon nanotubes, see [Ha], [SDD]) in a uniform magnetic field B = B(0, 0, 1) ∈ R3, B ∈ R. The corresponding
Dispersion for schrödinger equation with periodic potential in 1d
, 2007
"... Abstract. We extend a result on dispersion for solutions of the linear Schrödinger equation, proved by Firsova for operators with finitely many energy bands only, to the case of smooth potentials in 1D with infinitely many bands. The proof consists in an application of the method of stationary phase ..."
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Cited by 6 (1 self)
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Abstract. We extend a result on dispersion for solutions of the linear Schrödinger equation, proved by Firsova for operators with finitely many energy bands only, to the case of smooth potentials in 1D with infinitely many bands. The proof consists in an application of the method of stationary phase. Estimates for the phases, essentially the band functions, follow from work by Korotyaev. Most of the paper is devoted to bounds for the Bloch functions. For these bounds we need a detailed analysis of the quasimomentum function and the uniformization of the inverse of the quasimomentum function. §1
Conformal spectral theory for the monodromy matrix
, 2008
"... For any N ×N monodromy matrix we define the Lyapunov function, which is analytic on an associated Nsheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, whic ..."
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Cited by 6 (5 self)
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For any N ×N monodromy matrix we define the Lyapunov function, which is analytic on an associated Nsheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, antiperiodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (antiperiodic) eigenvalues or 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 imaginary part given by the Lyapunov exponent and we obtain various properties of this conformal mapping, which are similar to the case of the Hill operator, 3) we determine various new trace formulae for potentials and the Lyapunov exponent, 4) we obtain a priori estimates of gap lengths in terms of the Dirichlet integral. We apply these results to the Schrödinger operators and to first order periodic systems on the real line with a matrix valued complex selfadjoint periodic potential. 1
Spectral estimates for matrixvalued periodic Dirac operators, preprint 2006
"... We consider the first order periodic systems perturbed by a 2N × 2N matrixvalued periodic potential on the real line. 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 Nsheeted Rie ..."
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Cited by 6 (5 self)
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We consider the first order periodic systems perturbed by a 2N × 2N matrixvalued periodic potential on the real line. 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 Nsheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points, which we call resonances. We prove the existence of real or complex resonances. We determine the asymptotics of the periodic, antiperiodic 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 antiperiodic eigenvalues, ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). Moreover, we
Parametrization of Periodic Weighted Operators in Terms of Gap Lengths
, 1999
"... We consider the periodic weighted operator Ty = \Gammaae \Gamma2 (ae 2 y 0 ) 0 in L 2 (R; ae(x) 2 dx), where ae is a 1periodic real positive function with ae(0) = 1. We assume q = ae 0 =ae 2 L 2 (0; 1). The spectrum of T consists of intervals oe n = [ + n\Gamma1 ; \Gamma n ] ..."
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Cited by 5 (2 self)
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We consider the periodic weighted operator Ty = \Gammaae \Gamma2 (ae 2 y 0 ) 0 in L 2 (R; ae(x) 2 dx), where ae is a 1periodic real positive function with ae(0) = 1. We assume q = ae 0 =ae 2 L 2 (0; 1). The spectrum of T consists of intervals oe n = [ + n\Gamma1 ; \Gamma n ] separated by gaps ( \Gamma n ; + n ); n ? 1. Using essentially the square of the gap lengths, the centre of the gap and the Dirichlet eigenvalues on the unit interval we construct a gap length mapping q ! ` 2 \Phi ` 2 which provides a real analytic parametrization of the weight. For the proof we use nonlinear functional analysis in Hilbert space combined with sharp asymptotic estimates on the fundamental solution and the Lyapunov function in the high energy limit for complex q.