Discrete diffraction managed solitons: Threshold phenomena and rapid decay for general nonlinearities

We prove a threshold phenomenon for the existence/non-existence of energy minimizing solitary solutions of the diffraction management equation for strictly positive and zero average diffraction. Our methods allow for a large class of nonlinearities, they are, for example, allowed to change sign, and the weakest possible condition, it only has to be locally integrable, on the local diffraction profile. The solutions are found as minimizers of a nonlinear and nonlocal variational problem which is translation invariant. There exists a critical threshold ?cr such that minimizers for this variational problem exist if their power is bigger than ?cr and no minimizers exist with power less than the critical threshold. We also give simple criteria for the finiteness and strict positivity of the critical threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. Furthermore, we give precise quantitative lower bounds on the exponential decay rate of the diffraction management solitons, which confirm the physical heuristic prediction for the asymptotic decay rate. Moreover, for ground state solutions, these bounds give a quantitative lower bound for the divergence of the exponential decay rate in the limit of vanishing average diffraction. For zero average diffraction, we prove quantitative bounds which show that the solitons decay much faster than exponentially. Our results considerably extend and strengthen the results of [15] and [16].Comment: 49 pages, no figure

Exponential decay of eigenfunctions and generalized eigenfunctions of a non self-adjoint matrix Schr\"odinger operator related to NLS

We study the decay of eigenfunctions of the non self-adjoint matrix operator \calH = (\begin{smallmatrix} -\Delta +\mu+U & W \W & \Delta -\mu -U \end{smallmatrix}), for $\mu>0$, corresponding to eigenvalues in the strip -\mu<\re E <\mu.Comment: 16 page

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant symmetric \alpha-stable process in R^d and in an open set D respectively, where 0<\alpha < 2. The inequality G_D(x,y)G_D(y,z)/G_D(x,z) \le c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrodinger operators. When the above inequality is true with a constant c=c(D)>0, then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded \kappa-fat open set, which includes a bounded John domain. The 3G we consider is of the form G_D(x,y)G_D(z,w)/G_D(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form G_D(x,y)G_D(z,w)/G_D(x,w) appears in non-local Schrodinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on \kappa-fat open sets. As an application, we discuss relativistic \alpha-stable processes (relativistic Hamiltonian when \alpha=1) in \kappa-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic \alpha-stable processes in \kappa-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in \kappa-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Levy processes.Comment: 32 page

Absolutely Continuous Spectrum of a Polyharmonic Operator with a Limit Periodic Potential in Dimension Two

We consider a polyharmonic operator $H=(-\Delta)^l+V(x)$ in dimension two with $l\geq 6$, $l$ being an integer, and a limit-periodic potential $V(x)$. We prove that the spectrum contains a semiaxis of absolutely continuous spectrum.Comment: 33 pages, 8 figure

Spectral properties of a limit-periodic Schr\"odinger operator in dimension two

We study Schr\"{o}dinger operator $H=-\Delta+V(x)$ in dimension two, $V(x)$ being a limit-periodic potential. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle \vec k,\vec x\rangle }$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.Comment: 89 pages, 6 figure