The Absolutely Continuous Spectrum in One Dimension
The Absolutely Continuous Spectrum in One Dimension
复制标题
一维绝对连续谱
DOI：
--
复制
发表时间：
1983
期刊：
影响因子：
--
通讯作者：
B. Simon
中科院分区：
文献类型：
--
作者：
P. Deift;B. Simon
我们讨论了具有F概周期的H=-d/dx+V(X)的绝对连续谱及其离散模拟(Hu)(N)=u(n+1)+u(n-1)+V(Ri)u(Ri)。特别注意李亚波诺夫指数为零的能量集A。众所周知，这套设备是AC的基本支持。光谱测量的一部分。我们证明了A.E.为船体和A.E.放鳍。E在A，H和h中有连续本征函数，u9，u是概周期的。在离散情形中，我们证明了只有当V=const时，|^4|^4才相等。如果k是态的积分密度，我们证明了在A上，在连续情况下2kdk/de^π~，在离散情况下2πsmπkdk/de^.l。我们还给出了Pastur-Ishii定理和绝对连续谱的重数为2的新证明。
We discuss the absolutely continuous spectrum of H = — d/dx + V(x) with F almost periodic and its discrete analog (hu)(n) = u(n +1) + u(n — 1) + V(ri)u(ri). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure. We prove for a.e. Fin the hull and a.e. E in A, H and h have continuum eigenfunctions, u9 with u almost periodic. In the discrete case, we prove that |^4|^4 with equality only if V= const. If k is the integrated density of states, we prove that on A, 2kdk/dE^π~ in the continuum case and that 2πsmπkdk/dE^.l in the discrete case. We also provide a new proof of the Pastur-Ishii theorem and that the multiplicity of the absolutely continuous spectrum is 2.