On the symplectic self-adjointness and residual spectral emptiness of upper triangular Hamiltonian operator matrices

期刊论文
出版模式：

开放获取
发表日期：

2024年6月12日

查看全文

Jie Liu,
Guohai Jin,
Buhe Eerdun

摘要

This paper deals with the symplectic self-adjointness and residual spectral emptiness of upper triangular Hamiltonian operator matrices $H=\left( {\begin{matrix}A&{}B\\ 0&{}-A^*\end{matrix}}\right) $ $H = (\begin{matrix} A & B \\ 0 & - A^{*} \end{matrix})$ . First, for symplectic self-adjoint Hamiltonian operator H, based on detailed classification of point spectrum $\sigma _p(H)$ $σ_{p} (H)$ and residual spectrum $\sigma _r(H)$ $σ_{r} (H)$ , the symmetry about imaginary axis is given between $\sigma _p(H)$ $σ_{p} (H)$ , $\sigma _r(H)$ $σ_{r} (H)$ , deficiency spectrum $\sigma _{\delta }(H)$ $σ_{δ} (H)$ , compression spectrum $\sigma _\mathrm{{com}}(H)$ $σ_{com} (H)$ and approximate point spectrum $\sigma _\mathrm{{app}}(H)$ $σ_{app} (H)$ . Second, by means of the spectral symmetry, the sufficient and necessary conditions are given for $\sigma _r(H)=\varnothing $ $σ_{r} (H) = \emptyset$ , $\sigma _{r_1}(H)=\varnothing $ $σ_{r_{1}} (H) = \emptyset$ and $\sigma _{r_2}(H)=\varnothing $ $σ_{r_{2}} (H) = \emptyset$ , respectively. Then, for $H=\left( {\begin{matrix}A&{}B\\ 0&{}-A^*\end{matrix}}\right) $ $H = (\begin{matrix} A & B \\ 0 & - A^{*} \end{matrix})$ , it is proved that H is symplectic self-adjoint, if H is defined with diagonal domain ${\mathcal {D}}(H)={\mathcal {D}}(A)\oplus {\mathcal {D}}(A^*)$ $D (H) = D (A) \oplus D (A^{*})$ . Finally, for $H=\left( {\begin{matrix}A&{}B\\ 0&{}-A^*\end{matrix}}\right) $ $H = (\begin{matrix} A & B \\ 0 & - A^{*} \end{matrix})$ defined with diagonal domain, using the space decomposition, the sufficient and necessary conditions for $\sigma _r(H)=\varnothing $ $σ_{r} (H) = \emptyset$ and $\sigma _{r_1}(H)=\varnothing $ $σ_{r_{1}} (H) = \emptyset$ are described in detail, respectively, by line operator, null space, and range of inner elements.