Markov constant and quantum instabilities

Pelantová, Edita; Starosta, Štěpán; Znojil, Miloslav

doi:10.1088/1751-8113/49/15/155201

Cited by 4 publications

(6 citation statements)

References 65 publications

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“…Note that if we set r = 0 (except for n = 0) or r = a n+1 in (14), we get the convergents p n−1 q n−1 and p n+1 q n+1 , respectively. Let us resume well-known facts about values of convergents and semiconvergents:…”

Section: Continued Fractionsmentioning

confidence: 99%

One-sided Diophantine approximations

Hančl¹,

Turek²

2019

J. Phys. A: Math. Theor.

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The paper deals with best one-sided (lower or upper) Diophantine approximations of the -th kind ( ∈ N). We use the ordinary continued fraction expansions to formulate explicit criteria for a fraction p q ∈ Q to be a best lower or upper Diophantine approximation of the -th kind to a given α ∈ R. The sets of best lower and upper approximations are examined in terms of their cardinalities and metric properties. Applying our results in spectral analysis, we obtain an explanation for the rarity of so-called Bethe-Sommerfeld quantum graphs.

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Section: Continued Fractionsmentioning

confidence: 99%

One-sided Diophantine approximations

Hančl¹,

Turek²

2019

J. Phys. A: Math. Theor.

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“…, on the other hand, the literature on the Markov constant provides hints of the existence of numbers θ with the property υ(θ) = υ(θ −1 ), see e.g. [18]. holds for all p q = p q such that p , q ∈ Z and 0 < q ≤ q.…”

Section: Number Theoretic Preliminariesmentioning

confidence: 99%

“…For example, for the golden mean, φ = ( √ 5 + 1)/2, we have υ(φ) = υ(φ −1 ) = 1/ √ 5 (see Section 5 below), on the other hand, the literature on the Markov constant provides hints of the existence of numbers θ with the property υ(θ) = υ(θ −1 ), see e.g. [18]. Function υ(θ) is closely related to approximations of θ by rationals.…”

Section: Number Theoretic Preliminariesmentioning

confidence: 99%

Periodic quantum graphs from the Bethe–Sommerfeld perspective

Exner¹,

Turek²

2017

J. Phys. A: Math. Theor.

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Abstract. The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction is finite. To the date its validity is established for numerous systems, however, it is known that quantum graphs do not comply with this law as their spectra have typically infinitely many gaps, or no gaps at all. These facts gave rise to the question about the existence of quantum graphs with the 'Bethe-Sommerfeld property', that is, featuring a nonzero finite number of gaps in the spectrum. In this paper we prove that the said property is impossible for graphs with the vertex couplings which are either scale-invariant or associated to scale-invariant ones in a particular way. On the other hand, we demonstrate that quantum graphs with a finite number of open gaps do indeed exist. We illustrate this phenomenon on an example of a rectangular lattice with a δ coupling at the vertices and a suitable irrational ratio of the edges. Our result allows to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to the date.

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“…It is convenient to introduce a onesided analogue υ(θ) of the Markov constant, with the last inequality in (6) replaced by 0 < θ − p q < c q 2 . We have µ(θ) = min{υ(θ), υ(θ −1 )}; the number υ(θ) may or may not coincide with the Markov constant [21].…”

Section: Fig 1 the Rectangular-lattice Graphmentioning

confidence: 99%

Quantum graphs with the Bethe-Sommerfeld property

Exner¹,

Turek²

2017

Nanosystems: Phys. Chem. Math.

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In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned δ-coupling at the vertices.Keywords: periodic quantum graphs, gap number, δ-coupling, rectangular lattice graph, scale-invariant coupling, Bethe-Sommerfeld conjecture, golden mean.Quantum graphs [1] have attracted much attention both from the practical point of view as models of nanostructures as well as a tool to study the properties of quantum systems with a nontrivial topology of the configuration space. The topological richness of quantum graphs allows them to exhibit properties different from those of the 'usual' quantum Hamiltonians; examples are well known, for instance, the existence of compactly supported eigenfunctions on infinite graphs [1, Sec. 3.4] or the possibility of having flat bands only as is the case for magnetic chain graphs with a half-of-the-quantum flux through each chain element [2].In this letter, we are going to consider another situation where quantum graphs are known to behave unusually. Our problem concerns the gap structure of the spectrum of periodic quantum graphs. Recall that the finiteness of the open gap number for periodic quantum systems in dimension two or more was conjectured in the early days of quantum theory by Bethe and Sommerfeld [3]. The validity of the conjecture was taken for granted even if its proof turned out to pose a mathematically difficult problem. It took a long time before it was rigorously established for the 'usual' periodic Schrödinger operators [4][5][6][7][8]. Nevertheless, the situation appears to be different for quantum graphs, as we will see below.The traditional reasoning behind the Bethe-Sommerfeld conjecture relies on the behavior of the spectral bands identified with the ranges of the dispersion curves or surfaces which, in contrast to the one-dimensional situation, typically overlap, making opening of gaps more and more difficult as we proceed to higher energies. The situation with graphs might be similar [1, Sec. 4.7] but the spectral behavior need not be the same, one reason being the possibility of resonant gaps. The existence of gaps coming from a graph decoration was first observed in the discrete graph context [9] and the effect is present for metric graphs as well [1, Sec. 5.1]. In addition, the recently discovered universality property of periodic graphs [10], valid in the generic ...

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Markov constant and quantum instabilities

Cited by 4 publications

References 65 publications

One-sided Diophantine approximations

One-sided Diophantine approximations

Periodic quantum graphs from the Bethe–Sommerfeld perspective

Quantum graphs with the Bethe-Sommerfeld property

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