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We consider the Laplacian operator H_0 perturbed by a non-positive potential $V$, which is periodic in two directions, and decays in the remaining one. We are interested in the characterization and decay properties of the guided states, defined as the eigenfunctions of the reduced operators in the Bloch-Floquet-Gelfand transform of H_0+V in the periodic variables. If V is sufficiently small and decreases fast enough in the infinite direction, we prove that, generically, these guided states are characterized by quasi-momenta belonging to some one-dimensional compact real analytic submanifold of the Brillouin zone. Moreover they decay faster than any polynomial function in the infinite direction.
We investigate the dispersive properties of solutions to the Schrodinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schrodinger flow on each eigenspac
We prove regularity estimates for the eigenfunctions of Schrodinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual $N$-body operators are covered by our result; in tha
In this work we consider an example of a linear time-degenerate Schrodinger operator. We show that with the appropriate assumptions the operator satisfies a Kato smoothing effect. We also show that the solutions to the nonlinear initial value problem
For a periodic structure sandwiched between two homogeneous media, a bound state in the continuum (BIC) is a guided Bloch mode with a frequency in the radiation continuum. Optical BICs have found many applications, mainly because they give rise to re
We consider optimization problems for cost functionals which depend on the negative spectrum of Schrodinger operators of the form $-Delta+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under suitable ass