Spectral analysis of the one-speed transport operator and the functional model (Q1974244)

The functional model of Sz.-Nagy and Foiaş for dissipative operators is used to the spectral analysis of the operator \[ T= i\mu\partial_x+ ic(x) K,\quad Kf(x,\mu)= {1\over 2} \int^1_{-1} f(x,\mu') d\mu' \] acting in the Hilbert space \(H= L^2(\Gamma)\), \(\Gamma= \{(x,\mu); x\in\mathbb{R}, \mu\in[-1; 1]\}\). It is proved that if the function \(c\) belongs to the space \(L^+_0\) of essentially bounded, compactly supported and nonnegative almost everywhere functions, then the discrete spectrum of \(T\) consists of finitely many eigenvalues lying on the imaginary axis and its essential spectrum coincides with the real axis. The dimension of the subspace corresponding to the discrete spectrum is estimated. If \(T_{\text{ess}}\) is the restriction of \(T\) to the subspace corresponding to the essential spectrum, then the spectrum of \(T_{\text{ess}}\) is purely absolutely continuous. A singular set \(E\subset L^+_0\) is put into evidence such that if \(c\not\in E\), then \(T_{\text{ess}}\) is similar to a selfadjoint operator. For \(c\) in \(E\) the operator \(T_{\text{ess}}\) can be represented for sufficiently small \(\delta>0\) as a linear sum \(T_{\text{ess}}= T_1\dot+ T_2\) such that \(\sigma(T_1)= [-\delta; \delta]\) and \(T_2\) is similar to a selfadjoint operator whose spectrum is equal to the real axis.