Effective masses and conformal mappings

Kargaev, Pavel; Korotyaev, Evgeny

doi:10.1007/bf02099314

Cited by 45 publications

(78 citation statements)

References 8 publications

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“…That makes possible to reformulate the problem for the differential operator as a problem of the conformal mapping theory (see [KK1], [K1]- [K8] and [MO1]). We use the Poisson integral for the domain C + ∪ C − ∪ g n and some a priori estimates from [KK1]. The results of this paper are used in [KK3].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A priori estimates for the Hill and Dirac operators

Korotyaev

2008

Russ. J. Math. Phys.

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Consider the Hill operator T y = −y ′′ + q ′ (t)y in L 2 (R), where q ∈ L 2 (0, 1) is a 1-periodic real potential. The spectrum of T is is absolutely continuous and consists of bands separated by gaps γ n , n 1 with length |γ n | 0. We obtain a priori estimates of the gap lengths, effective masses, action variables for the KDV. For example, if µ ± n are the effective masses associated with the gap γ n = (λ − n , λ + n ), then |µ − n +µ + n | C|γ n | 2 n −4 for some constant C = C(q) and any n 1. In order prove these results we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping and the identities. Moreover, we obtain the similar estimates for the Dirac operator.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A priori estimates for the Hill and Dirac operators

Korotyaev

2008

Russ. J. Math. Phys.

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“…Then we should study the metric properties of a conformal mapping from C + onto a``comb'' K + . A similar analysis was done partially in [K1,K2,K3,KK1,KK2,KK33]. In the present paper we use an approach based on the identities for the Dirichlet integral (1.2) from [K2] and the embedding theorems (see Theorem 3.2).…”

Section: Introductionmentioning

confidence: 84%

“…The proof in [GT2] was not complete since the problem of an estimate of &q& in terms of &#& remained open. Kargaev and the author [KK1] found the estimate &#& 16 &q& and estimates of the gap lengths in terms of effective masses for the Dirac operator. In their next paper [KK2], Kargaev and the author reproved the result of Garnet and Trubowitz [GT1,GT2] by the direct method.…”

Section: Introductionmentioning

confidence: 98%

Estimates for the Hill Operator, I

Korotyaev

2000

Journal of Differential Equations

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“…which gives (1.22) We recall the result from [KK2]. Let a function f be harmonic and positive in the domain…”

Section: Theorem (Nevanlinna) I) Let μ Be a Borel Measure On R Such mentioning

confidence: 99%

“…Second, we obtain various results from the conformal mapping theory. For solving these "new" problems we use some techniques from [KK1], [KK2], [K2], [K6], [K7], [K8] and [CK], [K3]. In particular, we use the Poisson integral for the domain C + ∪ (−1, 1) ∪ C − .…”

Section: Introductionmentioning

confidence: 99%

Conformal spectral theory for the monodromy matrix

Korotyaev

2010

Trans. Amer. Math. Soc.

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Abstract. For any N × N monodromy matrix 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 Hill operator. 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. We show that the endpoints of each gap are periodic (anti-periodic) 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 self-adjoint periodic potential.

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Effective masses and conformal mappings

Cited by 45 publications

References 8 publications

A priori estimates for the Hill and Dirac operators

A priori estimates for the Hill and Dirac operators

Estimates for the Hill Operator, I

Conformal spectral theory for the monodromy matrix

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