18 results on '"Jeffrey Galkowski"'
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2. Semiclassical Resolvent Bounds for Long-Range Lipschitz Potentials
- Author
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Jacob Shapiro and Jeffrey Galkowski
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General Mathematics ,Operator (physics) ,010102 general mathematics ,Dimension (graph theory) ,Semiclassical physics ,Lipschitz continuity ,01 natural sciences ,Norm (mathematics) ,0103 physical sciences ,Elementary proof ,010307 mathematical physics ,Ball (mathematics) ,0101 mathematics ,Mathematics ,Resolvent ,Mathematical physics - Abstract
We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E> 0$. The potential is real valued and $V$ and $\partial _r V$ exhibit long-range decay at infinity and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C> 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.
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- 2021
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3. Analytic hypoellipticity of Keldysh operators
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Maciej Zworski and Jeffrey Galkowski
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Conjecture ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Neighbourhood (graph theory) ,Order (ring theory) ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Scattering theory ,0101 mathematics ,Lagrangian ,Analysis of PDEs (math.AP) ,Mathematical physics ,Mathematics - Abstract
We consider Keldysh-type operators, $ P = x_1 D_{x_1}^2 + a (x) D_{x_1} + Q (x, D_{x'} ) $, $ x = ( x_1, x') $ with analytic coefficients, and with $ Q ( x, D_{x'} ) $ second order, principally real and elliptic in $ D_{x'} $ for $ x $ near zero. We show that if $ P u =f $, $ u \in C^\infty $, and $ f $ is analytic in a neighbourhood of $ 0 $ then $ u $ is analytic in a neighbourhood of $ 0 $. This is a consequence of a microlocal result valid for operators of any order with Lagrangian radial sets. Our result proves a generalized version of a conjecture made by the second author and Lebeau and has applications to scattering theory., 24 pages, 1 figure
- Published
- 2021
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4. Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems
- Author
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John A. Toth and Jeffrey Galkowski
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Pointwise ,Polynomial (hyperelastic model) ,Physics ,010102 general mathematics ,Dimension (graph theory) ,Statistical and Nonlinear Physics ,Eigenfunction ,Riemannian manifold ,01 natural sciences ,Article ,Mathematics - Spectral Theory ,Combinatorics ,Projection (relational algebra) ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,Exponent ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Spectral Theory (math.SP) ,Mathematical Physics ,Analysis of PDEs (math.AP) - Abstract
Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2\Delta_g+V(x)-E_1$ so that $dp_1\neq 0$ on $p_1=0$. We assume that $P_1$ is quantum completely integrable in the sense that there exist functionally independent pseuodifferential operators $P_2,\dots P_n$ with $[P_i,P_j]=0$, $i,j=1,\dots ,n$. We study the pointwise bounds for the joint eigenfunctions, $u_h$ of the system $\{P_i\}_{i=1}^n$ with $P_1u_h=E_1u_h+o(1)$. We first give polynomial improvements over the standard H\"ormander bounds for typical points in $M$. In two and three dimensions, these estimates agree with the Hardy exponent $h^{-\frac{1-n}{4}}$ and in higher dimensions we obtain a gain of $h^{\frac{1}{2}}$ over the H\"ormander bound. In our second main result, under a real-analyticity assumption on the QCI system, we give exponential decay estimates for joint eigenfunctions at points outside the projection of invariant Lagrangian tori; that is at points $x\in M$ in the "microlocally forbidden" region $p_1^{-1}(E_1)\cap \dots \cap p_n^{-1}(E_n)\cap T^*_xM=\emptyset.$ These bounds are sharp locally near the projection of the invariant tori., Comment: 30 pages, 1 figure
- Published
- 2020
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5. Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem
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Eike Hermann Müller, Euan A. Spence, and Jeffrey Galkowski
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Dirichlet problem ,Helmholtz equation ,Applied Mathematics ,Boundary (topology) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Integral equation ,Generalized minimal residual method ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Mathematics - Analysis of PDEs ,Helmholtz free energy ,FOS: Mathematics ,symbols ,Applied mathematics ,Mathematics - Numerical Analysis ,35J05, 35J25, 65N22, 65N38, 65R20 ,0101 mathematics ,Galerkin method ,Boundary element method ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber $k$ to have the error in the iterative solution bounded independently of $k$ as $k\rightarrow \infty$ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions as $k \rightarrow \infty$ when the obstacle is nontrapping., Version 3 of this submission has been split into Version 4 and arXiv:1807.09719
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- 2019
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6. Semiclassical resolvent bounds for compactly supported radial potentials
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Kiril Datchev, Jeffrey Galkowski, and Jacob Shapiro
- Subjects
Mathematics - Analysis of PDEs ,FOS: Mathematics ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(|x|) - E$ in dimension $n \ge 2$, where $h, \, E > 0$, and $V: [0, \infty) \to \mathbb{R}$ is $L^\infty$ and compactly supported. The weighted resolvent norm grows no faster than $\exp(Ch^{-1})$, while an exterior weighted norm grows $\sim h^{-1}$. We introduce a new method based on the Mellin transform to handle the two-dimensional case.
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- 2021
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7. Domains Without Dense Steklov Nodal Sets
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Jeffrey Galkowski and Oscar P. Bruno
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Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,Sigma ,Mathematics::Spectral Theory ,Eigenfunction ,01 natural sciences ,Omega ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Ball (mathematics) ,0101 mathematics ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$-Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ωin two-dimensional domains $$\Omega $$Ω. In particular, this paper presents a dense family $$\mathcal {A}$$A of simply-connected two-dimensional domains with analytic boundaries such that, for each $$\Omega \in \mathcal {A}$$Ω∈A, the nodal set of the eigenfunction $$\phi _{\sigma _j}$$ϕσj “is not dense at scale $$\sigma _j^{-1}$$σj-1”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains $$\Omega \in \mathcal {A}$$Ω∈A, the nodal sets of the eigenfunctions $$\phi _{\sigma _j}$$ϕσj associated with the eigenvalue $$\sigma _j$$σj have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $$\Omega \in \mathcal {A}$$Ω∈A there is a value $$r_1>0$$r1>0 such that for each j there is $$x_j\in \Omega $$xj∈Ω such that $$\phi _{\sigma _j}$$ϕσj does not vanish on the ball of radius $$r_1$$r1 around $$x_j$$xj.
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- 2020
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8. Lower bounds for Cauchy data on curves in a negatively curved surface
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Steve Zelditch and Jeffrey Galkowski
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Surface (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Process (computing) ,Cauchy distribution ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Eigenfunction ,01 natural sciences ,Upper and lower bounds ,Mathematics - Analysis of PDEs ,Hypersurface ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Observability ,0101 mathematics ,Algebra over a field ,Mathematical Physics ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.
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- 2020
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9. Eigenfunction scarring and improvements in L∞ bounds
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Jeffrey Galkowski and John A. Toth
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Numerical Analysis ,Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Sense (electronics) ,Mathematics::Spectral Theory ,Eigenfunction ,01 natural sciences ,Measure (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Analysis ,Mathematics - Abstract
We study the relationship between L∞ 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∞ 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.
- Published
- 2018
- Full Text
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10. Wavenumber-Explicit Regularity Estimates on the Acoustic Single- and Double-Layer Operators
- Author
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Euan A. Spence and Jeffrey Galkowski
- Subjects
Dirichlet problem ,Pure mathematics ,Algebra and Number Theory ,Helmholtz equation ,010102 general mathematics ,Boundary (topology) ,Semiclassical physics ,31B10, 31B25, 35J05, 35J25, 65R20 ,01 natural sciences ,Omega ,symbols.namesake ,Mathematics - Analysis of PDEs ,Helmholtz free energy ,0103 physical sciences ,FOS: Mathematics ,symbols ,010307 mathematical physics ,0101 mathematics ,Galerkin method ,Laplace operator ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$ is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in $L^2(\partial \Omega)$, of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ bounds can then be used in a wavenumber-explicit version of the classic compact-perturbation analysis of Galerkin discretisations of these second-kind equations; this is done in the companion paper [Galkowski, M\"uller, Spence, arXiv 1608.01035]., Comment: Version 3 of 1608.01035 has been split into Version 4 of that submission and this present submission
- Published
- 2019
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11. Eigenfunction Concentration via Geodesic Beams
- Author
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Yaiza Canzani and Jeffrey Galkowski
- Subjects
Geodesic ,General Mathematics ,FOS: Physical sciences ,Lambda ,01 natural sciences ,Mathematics - Spectral Theory ,Superposition principle ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematical Physics ,Mathematics ,Mathematical physics ,Physics ,Pointwise ,Laplace transform ,Applied Mathematics ,010102 general mathematics ,Mathematical Physics (math-ph) ,Radius ,Eigenfunction ,010307 mathematical physics ,Beam (structure) ,Analysis of PDEs (math.AP) - Abstract
In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists of controlling $\phi_\lambda(x)$ by decomposing $\phi_\lambda$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $\lambda^{-\frac{1}{2}}$. We control $\phi_\lambda(x)$ by the $L^2$-mass of $\phi_\lambda$ on each geodesic tube and derive a purely dynamical statement through which $\phi_\lambda(x)$ can be studied. In particular, we obtain estimates on $\phi_\lambda(x)$ by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that are. This approach allows for quantitative improvements, in terms of $T$, on the available bounds for $L^\infty$ norms, $L^p$ norms, pointwise Weyl laws, and averages over submanifolds., Comment: 61 pages, 2 figures. Improved exposition and includes new explanatory material in the introduction as well as an examples section (1.5) and a full section on comparison with previous work (1.6). Appendices A.1 (Index of notation) and B were also added
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- 2021
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12. Control From an Interior Hypersurface
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Matthieu Léautaud, Jeffrey Galkowski, Léautaud, Matthieu, Stanford University, Centre de Mathématiques Laurent Schwartz (CMLS), and Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Cauchy distribution ,Eigenfunction ,Riemannian manifold ,Wave equation ,01 natural sciences ,Controllability ,Mathematics - Spectral Theory ,Hypersurface ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Heat equation ,[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We consider a compact Riemannian manifold $M$ (possibly with boundary) and $\Sigma \subset M\setminus \partial M$ an interior hypersurface (possibly with boundary). We study observation and control from $\Sigma$ for both the wave and heat equations. For the wave equation, we prove controllability from $\Sigma$ in time $T$ under the assumption $(\mathcal{T}GCC)$ that all generalized bicharacteristics intersect $\Sigma$ transversally in the time interval $(0,T)$. For the heat equation we prove unconditional controllability from $\Sigma$. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on $\Sigma$ under $\mathcal{T}GCC$ and unconditional exponential lower bounds on such Cauchy data., Comment: 45 pages 1 figure
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- 2017
- Full Text
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13. Fractal Weyl laws and wave decay for general trapping
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Semyon Dyatlov, Jeffrey Galkowski, and Dyatlov, Semen
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Geodesic ,General Physics and Astronomy ,STRIPS ,01 natural sciences ,Upper and lower bounds ,law.invention ,Mathematics - Spectral Theory ,Mathematics - Analysis of PDEs ,Fractal ,law ,0103 physical sciences ,Euclidean geometry ,FOS: Mathematics ,0101 mathematics ,Exponential decay ,Spectral Theory (math.SP) ,Mathematical Physics ,Mathematical physics ,Mathematics ,Scattering ,Applied Mathematics ,010102 general mathematics ,Statistical and Nonlinear Physics ,Weyl law ,010307 mathematical physics ,Analysis of PDEs (math.AP) - Abstract
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., Comment: 36 pages, 5 figures; minor revisions
- Published
- 2017
- Full Text
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14. Arbitrarily small perturbations of Dirichlet Laplacians are quantum unique ergodic
- Author
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Sourav Chatterjee and Jeffrey Galkowski
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Pure mathematics ,01 natural sciences ,Dirichlet distribution ,Mathematics - Spectral Theory ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Ergodic theory ,0101 mathematics ,Quantum ,Spectral Theory (math.SP) ,Mathematical Physics ,Physics ,010102 general mathematics ,Probability (math.PR) ,Statistical and Nonlinear Physics ,Eigenfunction ,Mathematics::Spectral Theory ,Quantum chaos ,Weyl law ,Domain (ring theory) ,symbols ,Euclidean domain ,58J51, 81Q50, 35P20, 60J45 ,Geometry and Topology ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
Given an Euclidean domain with very mild regularity properties, we prove that there exist perturbations of the Dirichlet Laplacian of the form $-(I+S_\epsilon)\Delta$ with $\|S_\epsilon\|_{L^2\to L^2}\leq \epsilon$ whose high energy eigenfunctions are quantum uniquely ergodic (QUE). Moreover, if we impose stronger regularity on the domain, the same result holds with $\|S_\epsilon\|_{L^2\to H^\gamma}\leq \epsilon$ for $\gamma>0$ depending on the domain. We also give a proof of a local Weyl law for domains with rough boundaries., Comment: 32 pages
- Published
- 2016
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15. The $L^2$ Behavior of Eigenfunctions Near the Glancing Set
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Jeffrey Galkowski
- Subjects
Laplace transform ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Scale (descriptive set theory) ,Eigenfunction ,01 natural sciences ,Manifold ,Mathematics - Spectral Theory ,Mathematics - Analysis of PDEs ,Hypersurface ,0103 physical sciences ,FOS: Mathematics ,Uniform boundedness ,Ergodic theory ,010307 mathematical physics ,0101 mathematics ,Spectral Theory (math.SP) ,Analysis ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
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 $(-h^2\Delta_g-1)u=0$ 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 $(1+h^2\Delta_H)_+^{s_0}u|_H$ 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., Comment: 26 pages, 1 figure
- Published
- 2016
- Full Text
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16. Resonances for thin barriers on the circle
- Author
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Jeffrey Galkowski
- Subjects
Statistics and Probability ,Logarithm ,General Physics and Astronomy ,02 engineering and technology ,STRIPS ,01 natural sciences ,Upper and lower bounds ,law.invention ,symbols.namesake ,Mathematics - Analysis of PDEs ,law ,Quantum mechanics ,FOS: Mathematics ,0101 mathematics ,Quantum ,Mathematical Physics ,Physics ,010102 general mathematics ,Resonance ,Statistical and Nonlinear Physics ,Frequency dependence ,021001 nanoscience & nanotechnology ,Modeling and Simulation ,Quantum graph ,symbols ,0210 nano-technology ,Hamiltonian (quantum mechanics) ,Analysis of PDEs (math.AP) - Abstract
We study high energy resonances for the operator $-\Delta_{V,\partial\Omega}:=-\Delta+\delta_{\partial\Omega}\otimes V $ when $V$ has strong frequency dependence. The operator $-\Delta_{V,\partial\Omega}$ 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 $\Omega=B(0,1)\subset \mathbb{R}^2$ and $V\equiv h^{-\alpha }V_0>0$ with $\alpha\leq 1$. Here $h^{-1}\sim \Re \lambda$ is the frequency. We give sharp bounds on the size of resonance free regions for $\alpha\leq 1$ and the location of bands of resonances when $5/6\leq \alpha\leq 1$. Finally, we give a lower bound on the number of resonances in logarithmic size strips: $-M\log \Re \lambda\leq \Im \lambda \leq 0$., Comment: 23 pages, 6 figure
- Published
- 2016
- Full Text
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17. Nonlinear Instability in a Semiclassical Problem
- Author
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Jeffrey Galkowski
- Subjects
Nonlinear instability ,Physics ,Complex energy ,Operator (physics) ,010102 general mathematics ,Spectrum (functional analysis) ,Mathematical analysis ,Complex system ,Semiclassical physics ,Statistical and Nonlinear Physics ,01 natural sciences ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,Uniform boundedness ,0101 mathematics ,010306 general physics ,Nonlinear evolution ,Mathematical Physics ,Analysis of PDEs (math.AP) - Abstract
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.
- Published
- 2011
- Full Text
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18. Synthesis of size-controlled and shaped copper nanoparticles
- Author
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Derrick Mott, Lingyan Wang, Jin Luo, Chuan-Jian Zhong, and Jeffrey Galkowski
- Subjects
Coalescence (physics) ,Stereochemistry ,Melting temperature ,Dispersity ,chemistry.chemical_element ,Nanoparticle ,Surfaces and Interfaces ,Condensed Matter Physics ,Copper ,chemistry ,Chemical engineering ,Electrochemistry ,Molecule ,General Materials Science ,Amine gas treating ,Size dependence ,Spectroscopy - Abstract
The synthesis of stable, monodisperse, shaped copper nanoparticles has been difficult, partially because of copper's propensity for oxidation. This article reports the findings of an investigation of a synthetic route for the synthesis of size-controllable and potentially shape-controllable molecularly capped copper nanoparticles. The approach involved the manipulation of reaction temperature for the synthesis of copper nanoparticles in organic solvents in the presence of amine and acid capping agents. By manipulating the reaction temperature, this route has been demonstrated for the production of copper nanoparticles ranging from 5 to 25 nm. The size dependence of the melting temperature of copper nanoparticles, especially for surface melting, is believed to play an important role in interparticle coalescence, leading to size growth as the reaction temperature is increased. Control of the reaction temperature and capping molecules has also been demonstrated to produce copper nanoparticles with different shapes such as rods and cubes. The previously proposed combination of the selective formation of a seed precursor and a selective growth direction due to the preferential adsorption of capping agents on certain nanocrystal facets is believed to be responsible for shape formation by kinetically controlling the growth rates of crystal facets. The nanoparticles are characterized using TEM, XRD, and UV-visible techniques. A mechanistic consideration of the size control and shape formation is also discussed.
- Published
- 2007
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