Hans Christianson

Dispersive Estimates for Manifolds with One Trapped Orbit

For a large class of complete, non-compact Riemannian manifolds, (M, g), with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator: where ρ s (x) ∈ 𝒞∞(M) satisfies ρ s = ⟨ dist g (x,x 0) ⟩−s , , and V ∈ 𝒞∞(M), 0 ≤ V ≤ C satisfies |∇ V| ≤ C ⟨ dist(x,x 0) ⟩−1−δ for some δ > 0. From the local smoothing estimate, we deduce a family of Strichartz-type estimates, which are used to prove two well-posedness results for the nonlinear Schrödinger equation. As a second application, we prove the following sub-exponential local energy decay estimate for solutions to the wave equation when dim M = n ≥ 3 is odd and M is equal to ℝ n outside a compact set: where ψ ∈ 𝒞∞(M), ψ ≡ e −|x|2 outside a compact set.

QUANTUM MONODROMY AND NONCONCENTRATION NEAR A CLOSED SEMI-HYPERBOLIC ORBIT

For a large class of semiclassical operators P(h) ― z which includes Schrodinger operators on manifolds with boundary, we construct the Quantum Monodromy operator M(z) associated to a periodic orbit γ of the classical flow. Using estimates relating M(z) and P(h) — z, we prove semiclassical estimates for small complex perturbations of P(h) — z in the case γ is semi-hyperbolic. As our main application, we give logarithmic lower bounds on the mass of eigenfunctions away from semi-hyperbolic orbits of the associated classical flow. As a second application of the Monodromy Operator construction, we prove if γ is an elliptic orbit, then P(h) admits quasimodes which are well-localized near γ.

The critical group of a threshold graph

The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph. The structure of this group is a subtle isomorphism invariant that has received much attention recently, partly due to its relation to the graph Laplacian and chip-firing games. However, the group structure has been determined for relatively few classes of graphs. Based on computer evidence, we conjecture the exact group structure for a well-studied class of graphs having integer spectra, the threshold graphs, and prove this conjecture for the subclass which we call generic threshold graphs.

Strichartz Estimates for the Water-Wave Problem with Surface Tension

Strichartz-type estimates for one-dimensional surface water-waves under surface tension are studied, based on the formulation of the problem as a nonlinear dispersive equation. We establish a family of dispersion estimates on time scales depending on the size of the frequencies. We infer that a solution u of the dispersive equation we introduce satisfies local-in-time Strichartz estimates with loss in derivative: where C depends on T and on the norms of the H s -norm of the initial data. The proof uses the frequency analysis and semiclassical Strichartz estimates for the linealized water-wave operator.

Local smoothing for the Schrödinger equation with a prescribed loss

We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schrödinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss.

Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space

We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.

Cutoff resolvent estimates and the semilinear schrödinger equation

This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrodinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schrodinger equation.

Nonlinear Bound States on Weakly Homogeneous Spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.

Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by exp(CK |s| � ) in strips |Re s| ≤ K, whereis the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott-Ruelle resonances of this dynamical system). An up- per bound on the number of zeros in polynomial regions {|Re s| ≤ |Im s| � } is given, followed by weaker lower bound estimates in strips {Re s > −C, |Im s| ≤ r}, and logarithmic neighbourhoods {|Re s| ≤ �log |Im s|}. Recent numerical workof Strain-Zworski suggests the upper bounds in strips are optimal.

Classification of order preserving isomorphisms between algebras of semiclassical operators

Following the work of Duistermaat and Singer on isomorphisms of algebras of global pseudodifferential operators, we classify order preserving isomorphisms of algebras of microlocally defined semiclassical pseudodifferential operators. Specifically, we show that any such isomorphism is given by conjugation by , where is a microlocally elliptic semiclassical pseudodifferential operator and is a microlocally unitary -FIO associated to the graph of a local symplectic transformation.