TY - JOUR

T1 - On the bethe-sommerfeld conjecture for higher-order elliptic operators

AU - Shen, Zhongwei

PY - 2003/5

Y1 - 2003/5

N2 - We consider the elliptic operator P(D) + V in ℝd, d ≥ 2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2l with a homogeneous convex symbol P(ξ), and V is a real periodic function in L∞(ℝd). We show that the number of gaps in the spectrum of P(D) + V is finite if 4l > d + 1. If in addition, V is smooth and the convex hyper-surface {ξ ∈ ℝd: P(ξ) = 1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D) + V is finite, provided 8l > d + 3 and 9 ≥ d ≥ 2, or 4l > d - 3 and d ≥ 10.

AB - We consider the elliptic operator P(D) + V in ℝd, d ≥ 2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2l with a homogeneous convex symbol P(ξ), and V is a real periodic function in L∞(ℝd). We show that the number of gaps in the spectrum of P(D) + V is finite if 4l > d + 1. If in addition, V is smooth and the convex hyper-surface {ξ ∈ ℝd: P(ξ) = 1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D) + V is finite, provided 8l > d + 3 and 9 ≥ d ≥ 2, or 4l > d - 3 and d ≥ 10.

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U2 - 10.1007/s00208-003-0395-z

DO - 10.1007/s00208-003-0395-z

M3 - Article

AN - SCOPUS:0038130590

SN - 0025-5831

VL - 326

SP - 19

EP - 41

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1

ER -