Diffusion bound and reducibility for discrete Schrodinger equations with tangent potential

作者：Zhang Shiwen^*; Zhao Zhiyan

来源：Frontiers of Mathematics in China, 2012, 7(6): 1213-1235.

DOI：10.1007/s11464-012-0241-2

• Tangent potential
• reducibility
• Sobolev norm
• Birkhoff normal form

In this paper, we consider the lattice Schrodinger equations
iq(n)(t) = tan pi(n alpha + x)q(n)(t) + epsilon(q(n+1)(t) + q(n-1)(t)) + delta nu(n)(t)vertical bar q(n)(t)vertical bar(2 tau)-(2)q(n)(t),
with alpha satisfying a certain Diophantine condition, x is an element of R/Z, and tau = 1 or where nu(n) (t) is a spatial localized real bounded potential satisfying vertical bar nu(n)(t)vertical bar <= Cc(-rho vertical bar n vertical bar). We prove that the growth of H-1 norm of the solution {q(n)(t)}(n is an element of Z) is at most logarithmic if the initial data {q(n) (0)}(n is an element of Z) is an element of H-1 for epsilon sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e.,
iq(n)(t) = tan pi(n alpha + x)q(n)(t) + epsilon(q(n+1)(t) + q(n-1)(t)) + delta nu(n)(theta(0) + t omega)q(n)(t). Then the linear equation can be reduced to an autonomous equation for a.e. x and most values of the frequency vectors omega if epsilon and delta are sufficiently small.

• 出版日期2012-12
• 单位南京大学