Solvability of boundary value problems for a Schrödinger-type operator-differential equation with varying time direction (Q2760722)

Let \(H\) be a separable complex Hilbert space, let \(L\) be a positive definite selfadjoint operator with dense domain \(D(L)\) in \(H\), and let \(B\) be a selfadjoint operator on \(H\). Denote by \(E^+\), \(E^0\), and \(E^-\) the spectral projections of \(B\) corresponding to the positive, zero, and negative parts of the spectrum. The author studies the following problems:NEWLINENEWLINENEWLINEProblem I. Find a solution to the equation NEWLINE\[NEWLINE Bu_t+iLu=Bf(t),\quad t\in[0,T] \quad (i^2=-1),NEWLINE\]NEWLINE satisfying NEWLINE\[NEWLINE E^+(u(0)-u_0)=0,\quad E^-(u(T)-u_T)=0, NEWLINE\]NEWLINE where \(u_0\) and \(u_T\) are given elements.NEWLINENEWLINENEWLINEProblem II. Find a solution to the equation NEWLINE\[NEWLINE Bu_t+iLu=Bf(t) NEWLINE\]NEWLINE satisfying NEWLINE\[NEWLINE u(0)-\mu u(T)=u_0 NEWLINE\]NEWLINE (with \(u_0\) a given element).NEWLINENEWLINENEWLINEFor the two problems, the author proves existence and uniqueness theorems.