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 -