arXiv subjects
Publications and source records attributed to Jeffrey Galkowski.
Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{ ϕ_h \}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $μ$. Let $H$ be a smooth hypersurface. Our main result says that when $μ$ is $\textit{not}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h \to 0^+$.
We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal $L^\infty$ growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.
We consider a compact Riemannian manifold $M$ (possibly with boundary) and $Σ\subset M\setminus \partial M$ an interior hypersurface (possibly with boundary). We study observation and control from $Σ$ for both the wave and heat equations. For the wave equation, we prove controllability from $Σ$ in time $T$ under the assumption $(\mathcal{T}GCC)$ that all generalized bicharacteristics intersect $Σ$ transversally in the time interval $(0,T)$. For the heat equation we prove unconditional controllability from $Σ$. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on $Σ$ under $\mathcal{T}GCC$ and unconditional exponential lower bounds on such Cauchy data.
We prove a Weyl upper bound on the number of scattering resonances in strips for manifolds with Euclidean infinite ends. In contrast with previous results, we do not make any strong structural assumptions on the geodesic flow on the trapped set (such as hyperbolicity) and instead use propagation statements up to the Ehrenfest time. By a similar method we prove a decay statement with high probability for linear waves with random initial data. The latter statement is related heuristically to the Weyl upper bound. For geodesic flows with positive escape rate, we obtain a power improvement over the trivial Weyl bound and exponential decay up to twice the Ehrenfest time.
Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{ϕ_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2Δ_gϕ_h=ϕ_h$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(\{ϕ_h\},H)$ for which $$ \Big|\int_Hϕ_hdσ_H\Big|=o(h^{\frac{1-k}{2}}),\qquad h\to 0^+. $$ One such condition is that the set of conormal directions to $H$ that are recurrent has measure $0$. In particular, we show that the upper bound holds for any $H$ if $(M,g)$ is surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.
We prove a quantum version of the Sabine law from acoustics describing the location of resonances in transmission problems. This work extends the author's previous work to a broader class of systems. Our main applications are to scattering by transparent obstacles, scattering by highly frequency dependent delta potentials, and boundary stabilized wave equations. We give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances or generalized eigenvalues to the chord lengths and reflectivity coefficients for the ray dynamics, thus proving a quantum version of the Sabine law.
Given an Euclidean domain with very mild regularity properties, we prove that there exist perturbations of the Dirichlet Laplacian of the form $-(I+S_ε)Δ$ with $\|S_ε\|_{L^2\to L^2}\leq ε$ whose high energy eigenfunctions are quantum uniquely ergodic (QUE). Moreover, if we impose stronger regularity on the domain, the same result holds with $\|S_ε\|_{L^2\to H^γ}\leq ε$ for $γ>0$ depending on the domain. We also give a proof of a local Weyl law for domains with rough boundaries.
Let $M$ be a compact manifold with or without boundary and $H\subset M$ be a smooth, interior hypersurface. We study the restriction of Laplace eigenfunctions solving $(-h^2Δ_g-1)u=0$ to $H$. In particular, we study the degeneration of $u|_H$ as one microlocally approaches the glancing set by finding the optimal power $s_0$ so that $(1+h^2Δ_H)_+^{s_0}u|_H$ remains uniformly bounded in $L^2(H)$ as $h\to 0$. Moreover, we show that this bound is saturated at every $h$-dependent scale near glancing using examples on the disk and sphere. We give an application of our estimates to quantum ergodic restriction theorems.
We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $τ$ with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate $τ$. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate $τ$, then there are resonances in logarithmic strips whose width is given by $τ$. As our main application of these results, we give sharp bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points. Moreover, these bounds are generically optimal on exteriors of nontrapping polygonal domains.
We study high energy resonances for the operator $-Δ_{V,\partialΩ}:=-Δ+δ_{\partialΩ}\otimes V $ when $V$ has strong frequency dependence. The operator $-Δ_{V,\partialΩ}$ is a Hamiltonian used to model both quantum corrals and leaky quantum graphs. Since highly frequency dependent delta potentials are out of reach of the more general techniques in previous work, we study the special case where $Ω=B(0,1)\subset \mathbb{R}^2$ and $V\equiv h^{-α}V_0>0$ with $α\leq 1$. Here $h^{-1}\sim \Re λ$ is the frequency. We give sharp bounds on the size of resonance free regions for $α\leq 1$ and the location of bands of resonances when $5/6\leq α\leq 1$. Finally, we give a lower bound on the number of resonances in logarithmic size strips: $-M\log \Re λ\leq \Im λ\leq 0$.
We study high energy resonances for the operators $-Δ+δ_{\partialΩ}\otimes V$ and $-Δ+δ_{\partialΩ}'\otimes V\partial_ν$ where $Ω$ is strictly convex with smooth boundary, $V:L^2(\partialΩ)\to L^2(\partialΩ)$ may depend on frequency, and $δ_{\partialΩ}$ is the surface measure on $\partialΩ$. These operators are model Hamiltonians for quantum corrals and leaky quantum graphs. We give a quantum version of the Sabine Law from the study of acoustics for both the $δ$ and $δ'$ interactions. It characterizes the decay rates (imaginary parts of resonances) in terms of the system's ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier. For the $δ$ interaction we show that generically there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of the $δ'$ interaction, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate and is optimal in the case of the unit disk in the plane. As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster. The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose--Taylor parametrix for complex energies. We use these constructions to give a complete microlocal description of boundary layer operators and to prove sharp high energy estimates on the boundary layer operators in the case that $\partialΩ$ is smooth and strictly convex.
In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency $λ^2\to\infty$. For single layer potentials, we find that the $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ norms decay in $λ$. The rate of decay depends on the curvature of $\partialΩ$: The norm is $λ^{-3/4}$ in general domains and $λ^{-5/6}$ if the boundary $\partialΩ$ is curved. The double layer potential, however, displays uniform $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. The appendix by Galkowski gives bounds $L^{2}(\partialΩ)\to{}L^{2}(\partialΩ)$ for the single and double layer operators at high frequency that are sharp modulo $\log λ$. In this case, both the single and double layer operator bounds depend upon the curvature of the boundary.
We consider operators $-Δ+ X$ where $X$ is a constant vector field, in a bounded domain and show spectral instability when the domain is expanded by scaling. More generally, we consider semiclassical elliptic boundary value problems which exhibit spectral instability for small values of the semiclassical parameter h, which should be thought of as the reciprocal of the Peclet constant. This instability is due to the presence of the boundary: just as in the case of $-Δ+ X$, some of our operators are normal when considered on R^d. We characterize the semiclassical pseudospectrum of such problems as well as the areas of concentration of quasimodes. As an application, we prove a result about exit times for diffusion processes in bounded domains. We also demonstrate instability for a class of spectrally stable nonlinear evolution problems that are associated to these elliptic operators.
We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the Hamiltonian flow.) Specificially, we assume that the billiard flow has an invariant ergodic component, U, and study defect measures, mu, of positivie density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence, when restricted to U, satisfies mu = c mu_L where mu_L is the Liouville measure. This proves part of a conjecture of Percival.
We establish high energy $L^2$ estimates for the restriction of the free Green's function to hypersurfaces in $\mathbb{R}^d$. As an application, we estimate the size of a logarithmic resonance free region for scattering by potentials of the form $V\otimes δ_Γ$, where $Γ\subset \mathbb{R}^d$ is a finite union of compact subsets of embedded hypersurfaces. In odd dimensions we prove a resonance expansion for solutions to the wave equation with such a potential.
We consider a nonlinear evolution problem with an asymptotic parameter and construct examples in which the linearized operator has spectrum uniformly bounded away from Re z >= 0 (that is, the problem is spectrally stable), yet the nonlinear evolution blows up in short times for arbitrarily small initial data. We interpret the results in terms of semiclassical pseudospectrum of the linearized operator: despite having the spectrum in Re z < -c < 0, the resolvent of the linearized operator grows very quickly in parts of the region Re z > 0. We also illustrate the results numerically.