Asymptotic distribution of eigenvalues of Hill's equation with ...

ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF HILL’S EQUATION 7Therefore it holds that0=∫ π0= − 14n 2 ∫ π+∫ π0U n (x)(4nd + d 2 + Q(x)) dx0It is easy to see∫ x0{ sin 2nx2nUsing the condition{sin 2 2n(x − x 1 ) } (4nd + d 2 + Q(x))(4nd + d 2 + Q(x 1 )) dx 1 dx}(K2 y 1 )(x) − (cos 2nx)(K 2 y 2 )(x) (4nd + d 2 + Q(x)) dx.∫ π ∫ x0∫ π0∫ π ∫ x0sin 2 2n(x − x 1 ) dx 1 dx = π24 .Q(x) dx =0,wehave{sin 2 2n(x − x 1 ) } (Q(x 1 )+Q(x)) dx 1 dx =0,00∫ π ∫ xHence d satisfies∣ (4nd + d 2 ) 2 −|¯Q 4n | 2∣ ∣=16n 2∣ π 20∫ π00{sin 2 2n(x − x 1 ) } Q(x 1 )Q(x) dx 1 dx = − π24 | ¯Q 4n | 2 .{ }sin 2nx2n(K2 y 1 )(x) − (cos 2nx)(K 2 y 2 )(x) (4nd + d 2 + Q(x)) dx∣(≦ 16n2‖y1 ‖ L ∞π 2 ‖K‖2 L ∞ →L ∞ 2n≦ 16(π|4nd + d 2 | + ‖Q‖ L 1nπ 2 1 −‖K‖ L ∞ →L ∞≦ C ( )π|4nd + d 2 | + ‖Q‖ L 1 {(4nd + d 2 ) 2 +1 } .n(3.2)Since we have already known d = o(1), it holds) 3(π|4nd+ ‖y 2 ‖ L ∞)+ d 2 )| + ‖Q‖ L 1π|4nd + d 2 | + ‖Q‖ L 1 = o(n).Combining this and ¯Q 4n = o(1) (by Riemann-Lebesgue’s Lemma) with (3.2), weobtain the assertion of Theorem 1.3□