arXiv subjects
Publications and source records attributed to R. Killip.
We consider the defocusing energy-critical nonlinear Schr\"odinger equation with inverse-square potential $iu_t = -\Delta u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a>-\frac14 +\frac1{25}$. We also carry out the variational analysis needed to treat the focusing case.
We study the $L^p$-theory for the Schr\"odinger operator $\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal L_a)^{s/2}$ agree with those defined via $(-\Delta)^{s/2}$. We consider all regularities $0<s<2$. In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities associated to the operator $\mathcal L_a$.
We consider the focusing energy-critical nonlinear Schrödinger equation $iu_t+Δu = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\in I}\|\nabla u(t)\|_2<\|\nabla W\|_2$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle in \cite{Evian, kenig-merle} for spherically symmetric initial data and dimensions $d=3,4,5$.
We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behaviour of this system.
We describe an ensemble of (sparse) random matrices whose eigenvalues follow the Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature beta. Our approach combines elements from the theory of orthogonal polynomials on the unit circle with ideas from recent work of Dumitriu and Edelman. In particular, we resolve a question left open by them: find a tri-diagonal model for the Jacobi ensemble.
We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian--the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.