The Second Order Estimates for the Hill Operatore
نویسنده
چکیده
Let H be the Hill operator and let G n = (A ? n ; A + n) and M n ; n 1; be the corresponding gaps and the eeective masses for the Hill operator. Denote by F(E) the Lyapunov function for the Hill operator. Let n 2 A ? n ; A + n ] be such that F 0 (n) = 0 and h n 0 be the solution of the equation cosh h n = (?1) n F(n): We prove some identities for the eeective masses M n : Then we nd "second order estimates" with respect to jG n j for n ; h n ; M n (for example j n ? 1 2 (A ? n + A + n)j CjG n j 2 =n 2), and we get ones for more general cases (the Dirac operator with periodic coeecients etc.).
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