Spectral Estimations for Can nical Systems

Winkler, Henrik

doi:10.1002/1522-2616(200012)220:1<115::aid-mana115>3.0.co;2-i

Cited by 10 publications

(32 citation statements)

References 10 publications

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“…Moreover, the problem was posed to characterize all diagonal matrix-functions H such that the m-function satisfies (1.8). Some progress was made by H. Winkler [59,Theorem 4.2] and as one of the main results in this article we will solve this problem (see Corollary 3.2 and Theorem 3.14). Our approach can be seen as a development of the ideas in [33] and [6].…”

Section: Introductionmentioning

confidence: 95%

Spectral asymptotics for canonical systems

Eckhardt

Kostenko

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Based on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of Weyl-Titchmarsh and spectral functions of 2 × 2 first order canonical systems. Our results improve several classical results and solve open problems posed by previous authors. Furthermore, they are applied to radial Dirac and radial Schrödinger operators as well as to Krein strings and generalized indefinite strings.

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Section: Introductionmentioning

confidence: 95%

Spectral asymptotics for canonical systems

Eckhardt

Kostenko

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…As the last ingredient on a canonical system, we see that a special type of a singular interval is necessary, when its spectral measure is finite, based on [14] or [16].…”

Section: Finite Measures and Singular Intervalsmentioning

confidence: 99%

“…Theorem 2.1 (Theorem 2.1 [14] or Theorem 4.11 [16]). The m-function of a canonical system has the form…”

Section: Finite Measures and Singular Intervalsmentioning

confidence: 99%

“…For this, a topology on canonical systems is required, such that it accords with the local uniform convergence on Herglotz functions. Besides de Branges' own work [2] for this (see also Lemma 2.1 in [14]), we adapt the following convergence on canonical systems in [6]: for all continuous functions f = (f 1 , f 2 ) t with compact support of [0, ∞).…”

Section: Non-density Of Discrete Schrödinger M-functionsmentioning

confidence: 99%

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The m-functions of discrete Schrödinger operators are sparse compared to those for Jacobi operators

Hur

2018

Journal of Differential Equations

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We explore the sparsity of Weyl-Titchmarsh m-functions of discrete Schrödinger operators. Due to this, the set of their m-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral theory for discrete Schrödinger operators via their spectral measures should be difficult.To obtain the result, de Branges theory of canonical systems is applied to work on them, instead of Weyl-Titchmarsh m-functions.

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“…For Sturm-Liouville equations such estimates go back to, at least, [14], [2] and [3]; for Krein strings estimates for the principal Titchmarsh-Weyl coefficient were proved in [22] and [23]; for canonical systems some results were obtained in [41]. Estimates of the distribution function of the spectral measure are studied in, e.g.…”

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confidence: 99%

Estimates for the Weyl coefficient of a two-dimensional canonical system

Langer¹,

Pruckner²,

Woracek³

2021

Preprint

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For a two-dimensional canonical system y ′ (t) = zJH(t)y(t) on some interval (a, b) whose Hamiltonian H is a.e. positive semi-definite and which is regular at a and in the limit point case at b, denote by qH its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment H → qH is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions.We give upper and lower bounds for |qH (z)| and Im qH (z) when z tends to i∞ non-tangentially. These bounds depend on the Hamiltonian H near the left endpoint a and determine |qH (z)| up to universal multiplicative constants. We obtain that the growth of |qH (z)| is independent of the off-diagonal entries of H and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for "most" Hamiltonians also Im qH (z) is fully determined).We translate the asymptotic behaviour of qH to the behaviour of the spectral measure µH of H by means of Abelian-Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at ±∞ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm-Liouville equations.

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Spectral Estimations for Can nical Systems

Cited by 10 publications

References 10 publications

Spectral asymptotics for canonical systems

Spectral asymptotics for canonical systems

The m-functions of discrete Schrödinger operators are sparse compared to those for Jacobi operators

Estimates for the Weyl coefficient of a two-dimensional canonical system

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