Resonances Analysis and Optimization

Transcription

1 Resonances Analysis and Optimization Fadil Santosa School of Mathematics, U Minnesota Institute for Mathematics and its Applications (IMA)

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3 Resonance optimization Analysis of resonance A perturbative approach Discussion Outline Credits: Chiu-Yen Kao, Claremont McKenna College David Dobson, University of Utah Stephan Shipman, Louisiana State University Michael Weinstein, Columbia University Junshan Lin, Auburn University

4 Consider the wave equation Resonance with V 0 -L -a a 0 L

5 Consider initial value problem Waves will be trapped in the well, but because the wall thickness on each side is finite, energy will escape. Optimization problem: What keeping the waves in the well? does the best job of To answer this question we have to start by looking at resonances.

6 One way to solve this problem is to put it in a finite domain with radiation BC s Take the Fourier transform

7 This non-selfadjoint eigenvalue problem admits a countable set of nontrivial solutions. The function is called a quasi-normal mode, the eigenvalue is called a resonance. The complex eigenvalues have the property where. Note that for ; it is not physical.

8 Compare the spectrum of the case where resonances ( finite) to Im k Continuous spectrum for L Bound states Re k Resonances

9 Quasi-normal modes approximates the solution of the wave equation. Choose. Then there exist a constant such that holds for every and. See Tang and Zworski 2000

10 Long time behavior is captured by the resonance pair, where, and is the smallest damping coefficient. The behavior is described by So, optimization problem amounts to designing which has the smallest.

11 Practical resonance calculation Consider the 1-D Schrödinger equation With having compact support and a well near x = 0. We calculate resonance by considering where is the free-space Green s function, extending its definition to the complex. We look for at which the above yields a nontrivial.

12 Optimization Because of there are an infinite number of resonances, the optimization problem is awkward to solve. Instead we devise a continuation algorithm. We start with a resonance and follow the steepest descent direction to reduce the negative complex component of the resonance. We have to take care to avoid resonances crossing along the trajectory.

13 Wave equation examples

14 2D example Initial Q= Final Q=321.95

15 We know how to calculate bound states, i.e.. As L gets large, k should get close to the associated bound state, the eigenvalue when. Im k Continuous spectrum for L Bound states Re k Resonances

16 Example V(x) 4 Mode is symmetric ψ x k B = L = ψ(x)

17 Can we find a way to perturb off of a bound state when is large?

18 Resonance problem Consider a symmetric potential and a symmetric quasimode. satisfies where

19 Perturbation approach Let be a bound state with frequency associated with potential V (x) We write Note: the difference between the operators associated with V L x and V x is not compact.

20 Plugging into equation we get The radiation BC leads to

21 Define a solution operator as follows. Let Then

22 Using the radiation boundary condition at x = L We can show (Dobson-S.-Shipman-Weinstein) The above equation has a solution for κ in the neighborhood of 0 and there are positive constants C, α, L, such that for L > L, the resonance is k = k B + κ, and κ Ce 2α(L a)

23 Example V(x) 4 1 L

24 A proposal for approximating resonance We write

25 Solve for the unique solution, denoted by Evaluate Approximate by

26 A simple method to find resonance from bound state frequency k B. Finding k B is relatively easy linear eigenvalue problem. If you know all k B s, we can find all nearby resonances for finite L.

27 Results V(x) 4 1 L 1 Exact Approximate

28 Extending to 2D The results presented uses symmetry of and the boundary condition at, this allows us to define a solution operator to solve the problem. The radiation boundary condition is local.

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30 Bound states satisfy with and having finite energy. We want to compare and.

31 Key ideas 1. Compare and in a common domain. 2. After fixing a quasi-mode, choose that is in some sense close to it. Accomplish 1. by exploiting the fact that Take care of 2. by choosing for

32 Bound state problem in

33 Resonance problem in D-to-N map is calculated by using the radiation D-to- N map at, explicit solution for, and continuity at.

34 Now consider. It satisfies and Solvability condition Together with makes the above uniquely solvable.

35 Main result Lin-S. Let be a bound state frequency corresponding to the potential There exists a constant such that for any, the following holds for the resonance in the neighborhood of where, and is a positive constant independent of.

36 Sketch of proof Write the resonance condition as. Expand You show that And does not vanish. is exponentially small, To establish these, we need estimates for and.

37 Bessel function estimates For n 0, and x < y non-negative

38 Example Angle independent quasi-mode (n = 0) k B =

39 Perturbative calculation Use Substituting, we get

40 Example

41 Discussion Near bound state resonances are exponentially close to the bound states when the well wall is thick. These resonances can be calculated using a perturbative method.