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Schrodinger equations with time-dependent interactions are studied. We investigate how to define the Floquet Hamiltonian as a self-adjoint operator, when the interaction is singular in time or space. Using these results we establish the existence of a bounded propagator, by applying a result given by J Howland. [ABSTRACT FROM AUTHOR]

We study a nonlinear ordinary differential equation on the half-line, with the Dirichlet boundary condition at the origin. This equation arises when studying the local maxima of the sojourn time for a free quantum particle whose states belong to an adequate subspace of the unit sphere of the corresponding Hilbert space. We establish several results concerning the existence and asymptotic behavior of the solutions. [ABSTRACT FROM AUTHOR]

We study Schrödinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by d-couplings with a parameter \alpha\in{\bb R} . If the graph is 'straight', i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum with all the gaps open whenever a [?] 0. We consider a 'bending' deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling a and the 'bending angle' as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands. [ABSTRACT FROM AUTHOR]