University of WashingtonQuantum Chaos in Scattering TheoryMaciej ZworskiUC Berkeley10 April 2007In this talk I want to relate objects of classical chaoticdynamics such as trapped sets and topologicalpressure of hyperbolic flows to the distribution ofquantum resonances. The plan is toIn this talk I want to relate objects of classical chaoticdynamics such as trapped sets and topologicalpressure of hyperbolic flows to the distribution ofquantum resonances. The plan is to• introduce quantum resonances using simpleone-dimensional modelsIn this talk I want to relate objects of classical chaoticdynamics such as trapped sets and topologicalpressure of hyperbolic flows to the distribution ofquantum resonances. The plan is to• introduce quantum resonances using simpleone-dimensional models• show how to compute them in that setting andhow their distribution is related to dynamicsIn this talk I want to relate objects of classical chaoticdynamics such as trapped sets and topologicalpressure of hyperbolic flows to the distribution ofquantum resonances. The plan is to• introduce quantum resonances using simpleone-dimensional models• show how to compute them in that setting andhow their distribution is related to dynamics• describe simple models of chaotic scattering indimension twoIn this talk I want to relate objects of classical chaoticdynamics such as trapped sets and topologicalpressure of hyperbolic flows to the distribution ofquantum resonances. The plan is to• introduce quantum resonances using simpleone-dimensional models• show how to compute them in that setting andhow their distribution is related to dynamics• describe simple models of chaotic scattering indimension two• relate the dimension of the trapped set to thedensity of resonancesIn this talk I want to relate objects of classical chaoticdynamics such as trapped sets and topologicalpressure of hyperbolic flows to the distribution ofquantum resonances. The plan is to• introduce quantum resonances using simpleone-dimensional models• show how to compute them in that setting andhow their distribution is related to dynamics• describe simple models of chaotic scattering indimension two• relate the dimension of the trapped set to thedensity of resonances• relate the pressure to the quantum decay rate.Simplest model:V (x) ∈ R ,Simplest model:V (x) ∈ R , |V (x)| ≤ C ,Simplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0Simplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 forSimplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 for |x| > L ,Simplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 for |x| > L ,and the corresponding Schr¨odinger operator,Simplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 for |x| > L ,and the corresponding Schr¨odinger operator,HVSimplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 for |x| > L ,and the corresponding Schr¨odinger operator,HV=Simplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 for |x| > L ,and the corresponding Schr¨odinger operator,HV= −∂2x+ V (x) .Simplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 for |x| > L ,and the corresponding Schr¨odinger operator,HV= −∂2x+ V (x) .Quantum resonances are the poles of the resolvent ofthe meromorphic continuation of Green’s function ofSimplest model:V (x) ∈ R , |V (x)| ≤ C , V (x) = 0 for |x| > L ,and the corresponding Schr¨odinger operator,HV= −∂2x+ V (x) .Quantum resonances are the poles of the resolvent ofthe meromorphic continuation of Green’s function ofHV− λ2.This definition although very elegant is not veryintuitive. Resonances manifest themselves veryconcretely in wave expansions, peaks of the scatt eringcross sections, and phase shift transitions.This definition although very elegant is not veryintuitive. Resonances manifest themselves veryconcretely in wave expansions, peaks of the scatt eringcross sections, and phase shift transitions.But first let us compute resonances in real time usinga MATLAB code (Bindel 2006).This definition although very elegant is not veryintuitive. Resonances manifest themselves veryconcretely in wave expansions, peaks of the scatt eringcross sections, and phase shift transitions.But first let us compute resonances in real time usinga MATLAB code (Bindel 2006).www.cims.nyu.edu/∼dbindel/resonant1dThe next movie shows the change of resonances of−h2∆ + V (x) as we make h → 0 whereThe next movie shows the change of resonances of−h2∆ + V (x) as we make h → 0 whereThe next movie shows the change of resonances of−h2∆ + V (x) as we make h → 0 whereWe went from h = 1 to h = 1/7 in 49 steps.The next movie shows the change of resonances of−h2∆ + V (x) as we make h → 0 whereWe went from h = 1 to h = 1/7 in 49 steps.Please note that as h decreases the density ofresonances goes up.Before explaining how resonances are related to thelong time behaviour of scattered waves we discuss themore familiar case of eigenvalues and eigenfuctions.Before explaining how resonances are related to thelong time behaviour of scattered waves we discuss themore familiar case of eigenvalues and eigenfuctions.A vibrating string, is described using eigenvalues andeigefunctions:∞Xj=0cos(tλj)cjuj(x) +∞Xj=0λ−1jsin(tλj)djuj(x)(for simplicity we assumed there are no negativeeigenvalues which holds, say, for V ≥ 0)Before explaining how resonances are related to thelong time behaviour of scattered waves we discuss themore familiar case of eigenvalues and eigenfuctions.A vibrating string, is described using eigenvalues andeigefunctions:∞Xj=0cos(tλj)cjuj(x) +∞Xj=0λ−1jsin(tλj)djuj(x)(for simplicity we assumed there are no negativeeigenvalues which holds, say, for V ≥ 0)This decomposition in the basis of harmonic analysis,signal processing, and many other things.On the whole line resonances and resonant statesreplace eigenvalues and eigenfunctions:On the whole line resonances and resonant statesreplace eigenvalues and eigenfunctions:X−A<Im λj≤0e−itλjcjuj(x) + rA(t, x) , t → +∞ ,(for simplicity we assumed there are no negativeeigenvalues which holds, say, for V ≥ 0, and moreseriously, that resonances are simple)On the whole line resonances and resonant statesreplace eigenvalues and eigenfunctions:X−A<Im λj≤0e−itλjcjuj(x) + rA(t, x) , t → +∞ ,(for simplicity we assumed there are no negativeeigenvalues which holds, say, for V ≥ 0, and moreseriously, that resonances are simple)The error satisfies the following