Refined asymptotics of the spectral gap for the Mathieu operator

Anahtarcı, Berkay; Djakov, Plamen

doi:10.1016/j.jmaa.2012.06.019

Cited by 13 publications

(18 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But it is impossible to apply Theorem 11 if the weight Ω is growing so slowly that 1 |k| Ω(k) = ∞. For example, this is the case if we consider the weight Ω given by For such weights, the next theorem gives conditions which guarantee that V (±2n) is the main term in the asymptotics of β ± n .…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Riesz basis property of Hill operators with potentials in weighted spaces

Djakov¹,

Mityagin²

2014

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

Consider the Hill operator L(v) = −d 2 /dx 2 +v(x) on [0, π] with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough n close to n 2 there are one Dirichlet eigenvalue μ n and two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λ − n , λ + n (counted with multiplicity). We describe classes of complex potentials v(x) = 2Z V (k)e ikx in weighted spaces (defined in terms of the Fourier coefficients of v) such that the periodic (or antiperiodic) root function system of L(v) contains a Riesz basis if and only ifThe theory of self-adjoint ordinary differential operators (o.d.o.) is well-developed, and the spectral decompositions play a central role in it [24,29,26].Convergence of the spectral decompositions of non-self-adjoint o.d.o., considered on a finite interval I and subject to strictly regular boundary conditions (see [29, § 4.8]), has been understood completely in the early 1960's [27,23,17]. In this case, we not only have convergence, but the system of eigenfunctions (SEF) is a Riesz basis in L 2 (I). However, in the case of regular but not strictly regular boundary conditions-even in the case of periodic or antiperiodic boundary conditions-complete understanding appeared only in the 2000s as a result of the interaction of two lines of research.One stems from a question raised by A. Shkalikov in 1996A. Shkalikov in /1997 in the Kostyuchenko-Shkalikov seminar on Spectral Analysis at Moscow State University. Shkalikov formulated the following assertion and sketched an approach to its proof.Consider the Hill operatorwith a smooth potential q such that for some s ≥ 0 q (k) (0) = q (k) (π), 0 ≤ k ≤ s − 1,2010 Mathematics Subject Classification. Primary 47E05, 34L40, 34L10.

show abstract

Section: Resultsmentioning

confidence: 99%

“…We refer to [11,12,15,16] for results about the asymptotics of β ± n , γ n = λ + n − λ − n , δ n = µ n − λ + n and Riesz basis property of root function systems in the case of potentials that are trigonometric polynomials. See also [2,1] and the bibliography therein.…”

Section: Example 16mentioning

confidence: 99%

Riesz basis property of Hill operators with potentials in weighted spaces

Djakov¹,

Mityagin²

2014

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

show abstract

“…a j cos(2jt), Levy and Keller [28] (see also [4] for a different approach) proved that the length of the N -th resonance interval is at most C N q r , where r is the integer part of N/s, and presented explicit formulas for C N when N is a multiple of s (see also [23], and [37] for interesting extensions to a generalized Ince equation). For the similar, and partly related, problem of the asymptotics of L N as N → ∞, we refer to [6,2]. In this paper we prove the following theorem which shows that, for every equation 1.1 coupled with 1.2, the instability tongues have at least the same order of tangency of the Mathieu equation, that is L N (q) = O(q N ) as q → 0.…”

Section: Introductionmentioning

confidence: 83%

On the instability tongues of the Hill equation coupled with a conservative nonlinear oscillator

Marchionna

Panizzi²

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We study the asymptotics for the lengths LN (q) of the instability tongues of Hill equations that arise as iso-energetic linearization of two coupled oscillators around a single-mode periodic orbit. We show that for small energies, i.e. q → 0, the instability tongues have the same behavior that occurs in the case of the Mathieu equation: LN (q) = O(q N ). The result follows from a theorem which fully characterizes the class of Hill equations with the same asymptotic behavior. In addition, in some significant cases we characterize the shape of the instability tongues for small energies. Motivation of the paper stems from recent mathematical works on the theory of suspension bridges.

show abstract

“…Formulas (5) and (6) can be obtained from − Ψ ′′ n,t (x) + q(x)Ψ n,t = λ n (t)Ψ n,t (x) (7) by multiplying e i(2πn+t)x and e i(2π(n−k)+t)x respectively. The uniform asymptotic formulas for the operator L t (q),with q ∈ L 1 [0, 1], generated in L 2 [0, 1] by (1) and (3) is obtained in [9], where we proved the following:…”

mentioning

confidence: 85%

“…where a and b are complex numbers. It is well-known that (see [4,8]) the spectrum S(H(a, b)) of the operator H(a, b) is the union of the spectra S(H t (a, b)) of the operators H t (a, b) for t ∈ (−π, π], where H t (a, b) is the operator generated in L 2 [0, 1] by (1) with potential (2) and by the boundary conditions y(1) = e it y(0), y ′ (1) = e it y ′ (0).…”

mentioning

confidence: 99%

Isospectral Mathieu–Hill Operators

Veliev

2013

Lett Math Phys

View full text Add to dashboard Cite

In this paper we prove that the spectra of the Mathieu-Hill Operators with potentials ae −i2πx + be i2πx and ce −i2πx + de i2πx are the same if and only if ab = cd, where a, b, c and d are complex numbers. This implies some corollaries about the extension of Harrell-Avron-Simon formula. Moreover, we find explicit formulas for the eigenvalues and eigenfunctions of the t-periodic boundary value problem for the Hill operator with Gasymov's potential.

show abstract

Refined asymptotics of the spectral gap for the Mathieu operator

Cited by 13 publications

References 24 publications

Riesz basis property of Hill operators with potentials in weighted spaces

Riesz basis property of Hill operators with potentials in weighted spaces

On the instability tongues of the Hill equation coupled with a conservative nonlinear oscillator

Isospectral Mathieu–Hill Operators

Contact Info

Product

Resources

About