Generalized Eigenvectors Of Quantum Walks

One-dimensional two-state space-inhomogeneous quantum walks are also called the one-dimensional two-state position-dependent quantum walks,which are a kind of quantum walks with the square summation function space l2(Z,C2)as the state space.In the study of quantum walks model,the eigenvalues and eigenvectors of time evolution operators play a crucial role.In quantum mechanics,observations of quantum systems are represented by a self-adjoint operator in Hilbert space.When the observed(self-adjoint operator)has only a point spectrum,the corresponding eigenvector belongs to the corresponding Hilbert space,and when the observed(selfadjoint operator)has a continuous spectrum,there may be "Eigenvector" that does not belong to the corresponding Hilbert space,it’s called a generalized eigenvector(a generalized state).This paper studies a kind of space-inhomogeneous linear quantum walks and a kind of space-inhomogeneous nonlinear quantum walks on the state space l2(Z,C2),The main work is as follows:We start out with constructing a rigged Hilbert space S(?) l2(Z,C2)(?)S*of the space l2(Z,C2),where S is a continuous and dense nuclear space embedded in l2(Z,C2)and S*is the dual space of S,the generalized eigenvector problem of space-inhomogeneous linear quantum walks is studied in the framework of rigged Hilbert space,and the corresponding existence results and other related results are obtained.For a kind of space-inhomogeneous nonlinear quantum walks model with l2(Z,C2)as the state space,we use the rigged Hilbert space method and the transfer matrix method to discuss the generalized eigenvector problem and other related problems,and obtain the corresponding results.