Description of the spectral bands for some 2D periodic Schrödinger operators

Boumaza, Hakim; Lafitte, Olivier

doi:10.1109/dd.2016.7756818

Cited by 2 publications

(5 citation statements)

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“…After multiplication by h 2 3 , the first term in the estimate ( 10) is equal to −1 and corresponds to the minimum of the potential q defined in (5).…”

Section: Notations and Main Resultsmentioning

confidence: 99%

“…We set = e − 4 3 (c−a1) 3 2 ∈ (0, e − 4 3 τ 3 2 ] and Y = 1 X. Since α = v u (−a 1 ) and using the fact that v u = 1 u 2 is bounded from below by α 2 by (55), one shows with a similar proof as (72) that:…”

Section: (88)mentioning

confidence: 86%

“…(2) We hope to use our results on the spectrum of H to study other periodic operators at least in dimension 2. First results were presented in [3]. In this case, we were to tackle the case where our operator decompose into a tensor product of two periodic Airy-Schrödinger operators with possibly two different characteristic lengths L 0 and L 1 .…”

Section: Discussionmentioning

confidence: 99%

“…It shows that the width of the spectral bands (multiplied by h 2 3 to take in account the scale in h of the right member of ( 5)) is proportional to…”

Section: 3mentioning

confidence: 99%

“…Again, we set = e − 4 3 (c−ã1) 3 2 ∈ (0, e − 4 3 (c0−ã1) 3 2 ]. We also define Y as in the proof of point (2) of Theorem 2.…”

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confidence: 99%

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The band spectrum of the periodic Airy–Schrödinger operator on the real line

Boumaza

Lafitte

2018

Journal of Differential Equations

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We introduce the periodic Airy-Schrödinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a semiclassical regime in which the results are stated for a fixed value of the semiclassical parameter and are thus estimates instead of asymptotic results. We prove that there exists a sequence of explicit constants, which are zeroes of classical functions, giving upper bounds of the semiclassical parameter for which the spectral bands are in the semiclassical regime. We completely determine the behaviour of the edges of the first spectral band with respect to the semiclassical parameter. Then, we investigate the spectral bands and gaps situated in the range of the potential. We prove precise estimates on the widths of these spectral bands and these spectral gaps and we determine an upper bound on the integrated spectral density in this range. Finally, in the semiclassical regime, we get estimates of the edges of every spectral bands and thus of the widths of every spectral bands and spectral gaps.1991 Mathematics Subject Classification. 34C25, 34E05, 34L40, 81Q80.

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“…After multiplication by h 2 3 , the first term in the estimate ( 10) is equal to −1 and corresponds to the minimum of the potential q defined in (5).…”

Section: Notations and Main Resultsmentioning

confidence: 99%

Section: (88)mentioning

confidence: 86%

Section: Discussionmentioning

confidence: 99%

“…It shows that the width of the spectral bands (multiplied by h 2 3 to take in account the scale in h of the right member of ( 5)) is proportional to…”

Section: 3mentioning

confidence: 99%

“…Again, we set = e − 4 3 (c−ã1) 3 2 ∈ (0, e − 4 3 (c0−ã1) 3 2 ]. We also define Y as in the proof of point (2) of Theorem 2.…”

mentioning

confidence: 99%

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