Quantum dynamics under continuous projective measurements: Non-Hermitian description and the continuum-space limit

The problem of the time arrival a quantum system in specified state is considered framework repeated measurement protocol and particular limit continuous measurements discussed. It shown that for choice system-detector coupling, Zeno effect avoided can be described effectively by non-Hermitian effective Hamiltonian. As specific example we consider evolution particle on one-dimensional lattice subjected to position at site. By solving corresponding wave-function equation, present analytic closed-form results survival probability first distribution. Finally discuss vanishing spacing show this leads continuum description where evolves via free Schr\"odinger equation with complex Robin boundary conditions detector Several interesting physical dynamics are presented.