A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions
A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions
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解析性条件下操纵希尔伯特空间上线性算子的谱理论
DOI：
10.1016/j.aim.2015.01.001
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发表时间：
2011
影响因子：
1.7
通讯作者：
Hayato Chiba
中科院分区：
文献类型：
--
作者：
Hayato Chiba
在Hilbert空间H上的线性算子T是自伴算子的扰动，且自伴算子的谱测度在某种意义下在真实的轴附近有解析延拓的条件下，建立了操纵Hilbert空间（Gelfand三元组）上线性算子的谱理论.证明了存在H的稠密子空间X使得算子T的预解式（λ− T）− 1 λ对任意λ ∈ X都有从下半平面到上半平面的解析延拓作为X′-值全纯函数，即使T在R上有连续谱，其中X′是X的对偶空间。被操纵的希尔伯特空间由三个空间X <$H <$X′组成。利用预解式的解析延拓作为从X到X′的算子，定义了X′中的广义本征值和广义本征函数。通常的谱理论的其他基本工具，如谱，预解式，Riesz投影和半群也研究了在一个操纵希尔伯特空间。它们被证明具有与通常谱理论相同的性质。所得结果可用于估计发展方程解的渐近性态。
A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator T on a Hilbert space H is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace X of H such that the resolvent (λ− T)− 1 ϕ of the operator T has an analytic continuation from the lower half plane to the upper half plane as an X′-valued holomorphic function for any ϕ∈ X, even when T has a continuous spectrum on R, where X′ is a dual space of X. The rigged Hilbert space consists of three spaces X⊂ H⊂ X′. A generalized eigenvalue and a generalized eigenfunction in X′ are defined by using the analytic continuation of the resolvent as an operator from X into X′. Other basic tools of the usual spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.
精确可解的薛定谔算子
DOI：
10.1007/s00023-011-0077-4
发表时间：
2011
期刊：
Annales Henri Poincaré
影响因子：
--
作者：
J. Dereziński;M. Wrochna
通讯作者：
M. Wrochna