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Resonances: Interpretation, Computation, and Perturbation David Bindel Department of Computer Science Cornell University 14 March 2011

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• Resonances:Interpretation, Computation, and Perturbation

David Bindel

Department of Computer ScienceCornell University

14 March 2011

• The quantum corral

• Particle in a box

Time-harmonic Schrdinger equation:

H =( d

2

dx2+ V

) = E

where

V (x) =

{0, 0 < x < 1, otherwise.

L2 solutions exist for En = k2n = n22:

=

{sin(knx), 0 < x < 10, otherwise.

• Particle in a box 2

Time-harmonic Schrdinger equation:

H =( d

2

dx2+ V

) = E

where

V (x) =

{0, 0 < x < 1E, otherwise.

L2 solutions exist for discrete values below E. Have the form

=

A exp(x

E E), x 0

B sin(

Ex) + C cos(

Ex), 0 < x < 1D exp((1 x)

E E), x 0,

where and are continuous (four constraints).

• Particle in a box 3

Time-harmonic Schrdinger equation:

H =( d

2

dx2+ V

) = E

where

V (x) =

0, 0 < x < 1E, 1 < x < 1 + L and L < x < 00 otherwise.

No L2 solutions! But a two-parameter family of boundedscattering solutions for any E = k2 > 0.

• Electrons unbound

For a finite barrier, electrons can escape!Not a bound state (conventional eigenmode).

• Scattering solutions

Schrdinger scattering from a potential V on [a,b]

H =( d

2

dx2+ V

) = E

For E = k2 > 0, get solutions

= eikx + scatter

where scatter satisfies outgoing BCs:(ddx ik

) = 0, x = b(

ddx

+ ik) = 0, x = a,

or via a Dirichlet-to-Neumann (DtN) map: (n B(k)) = 0

• Spectra and scattering

For compactly supported V , spectrum consists ofI Possible discrete spectrum (bound states) in (,0)I Continuous spectrum (scattering states) in [0,)

Were interested in the latter.

• Resonances and scattering

1.0 1.5 2.0 2.5 3.0 3.5 4.0k

50

100

|(k

)|

For supp(V ) , consider a scattering experiment:

(H k2) = f on (n B(k)) = 0 on

A measurement (k) = w shows a resonance peaks.Associate with a resonance pole k C (Breit-Wigner):

(k) C(k k)1.

• Resonances and scattering

Consider a scattering measurement (k)I Morally looks like = w(H E)1f?I w(H E)1f is well-defined off spectrum of HI Continuous spectrum of H is a branch cut for I Resonance poles are on a second sheet of definition for I Resonance wave functions blow up exponentially (not L2)

• Resonances and transients

A thousand valleys rustling pines resound.My heart was cleansed, as if in flowing water.In bells of frost I heard the resonance die.

Li Bai (interpreted by Vikram Seth)

• Resonances and transients

!3 !2 !1 0 1 2 3 4 5

0

50

100

Potential

!20 !15 !10 !5 0 5 10 15 20!0.8

!0.6

!0.4

!0.2

0Pole locations

• Resonances and transients

Lavf52.31.0

outs.mp4Media File (video/mp4)

• Eigenvalues and resonances

Eigenvalues ResonancesPoles of resolvent Second-sheet poles of extended resolventVector in L2 Wave function goes exponentialStable states TransientsPurely real Imaginary part describes local decay

• Computing resonances 1: Prony

Simplest method: extract resonances from (k)I This is the (modified) Prony methodI Has been used experimentally and computationally

(e.g. Wei-Majda-Strauss, JCP 1988 modified Prony applied to time-domain simulations)

But this is numerically sensitive, may require long simulations.

• Computing resonances 2: complex scaling

Change coordinates to shift the branch cut:

H =( d

2

dx2+ V

) = E

where dx/dx = 1 + i(x) is deformed outside [a,b].

I Rotates the continuous spectrum to reveal resonancesI First used to define resonances (Simon 1979)I Also a computational method:

I Truncate to a finite x domain.I Discretize using standard methodsI Solve a complex symmetric eigenvalue problem

One of my favorite computational tactics.

• Computing resonances 3: a nonlinear eigenproblem

Can also define resonances via a NEP:

(H k2) = 0 on (n B(k)) = 0 on

Resonance solutions are stationary points with respect to of

(, k) =

[()T () + (V k2)

]d

B(k) d

Discretized equations (e.g. via finite or spectral elements) are

A(k) =(

K k2M C(k)) = 0

K and M are real symmetric and C(k) is complex symmetric.

I PronyI Relatively simple signal processingI Can be used with scattering experiment resultsI May require long simulationsI Numerically sensitive

I Complex scalingI Straightforward implementationI Yields a linear eigenvalue problemI How to choose scaling parameters, truncation?

I DtN map formulationI Bounded domain no artificial truncationI Yields a nonlinear eigenvalue problemI DtN map is spatially nonlocal except in 1D

(though diagonalized by Fourier modes on a circle)

• The 1D case: MatScat

http://www.cs.cornell.edu/~bindel/cims/matscat/

http://www.cs.cornell.edu/~bindel/cims/matscat/http://www.cs.cornell.edu/~bindel/cims/matscat/

• Basic MatScat strategy

Pseudospectral collocation at Chebyshev points:(D2 + V (x) k2

) = 0, x (a,b)

(D ik) = 0, x = b(D + ik) = 0, x = a

Convert to linear problem with auxiliary variable = k.

• Is it that easy?

400 200 0 200 40040

30

20

10

0

10

20

All eigenvaluesChecked eigenvalues

• Is it that easy?

10 8 6 4 2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2Potential

2 1.5 1 0.5 0 0.5 1 1.5 22

1.5

1

0.5

0Pole locations

• Computation isnt enough!

Desired features:I A method to compute all resonances in some regionI and some sense of the accuracy of the computationI and some notion of sensitivity of the problem

• Steps toward sensitivity

Resonance solutions are stationary points with respect to of

(, k) =[2 + (V k2)

]d

(

n B(k)

)d

=

[()T () + (V k2)

]d

B(k) d

If (, k) a resonance pair, then (, k) = 0 and D(, k) = 0.

• Potential perturbations

If (, k) a resonance pair, then (, k) = 0 and D(, k) = 0.What if we perturb V?

= D + DV V + Dk k = 0

Note that D = 0! So

k = DV VDk

• Perturbation worked out

So look at how perturbations V change k :

k =

V

2

2k

2 B

(k)

Can also write in terms of a residual for as a solution for thepotential V + V :

k =

( + (V + V ) k

2)

2k

2 B

(k).

• Backward error analysis in MatScat

1. Compute approximate solution (, k).2. Map to high-resolution quadrature grid to evaluate

k =

( + V k

2)

2k

2 B

(k).

3. If k large, discard k as spurious; otherwise, acceptk k + k .

• But...

I Solving the 1D problem was only easy because it turnedinto a quadratic eigenvalue problem.

I In higher dimensions, get a more general nonlineareigenvalue problem.

I Can I combine a linear eigenvalue problem with erroranalysis worked out using the DtN map?

• Linear eigenproblems

Can also compute resonances byI Adding a complex absorbing potentialI Complex scaling methods

Both result in complex-symmetric ordinary eigenproblems:

(Kext k2Mext )ext =([

K11 K12K21 K22

] k2

[M11 M12M21 M22

])[12

]= 0

where 2 correspond to extra variables (outside ).

• Spectral Schur complement

0 5 10 15 20 25 30

0

5

10

15

20

25

30

0 5 10 15 20 25 30

0

5

10

15

20

25

30

Eliminate extra variables 2 to get

A(k)1 =(

K11 k2M11 C(k))1 = 0

where

C(k) = (K12 k2M12)(K22 k2M22)1(K21 k2M21)

• Aside on spectral Schur complement

Inverse of a Schur complement is a submatrix of an inverse:

(Kext z2Mext )1 =[A(z)1

]So for reasonable norms,

A(z)1 (Kext z2Mext )1.

Or

(A) (Kext ,Mext ),

(A) {z : A(z)1 > 1}(Kext ,Mext ) {z : (Kext z2Mext )1 > 1}

• Apples to oranges?

A(k) = (K k2M C(k)) = 0 (exact DtN map)

A(k) = (K k2M C(k)) = 0 (spectral Schur complement)

Two ideas:I Perturbation theory for NEP for local refinementI Complex analysis to get more global analysis

• Linear vs nonlinear

-5

-4

-3

-2

-1

0

0 2 4 6 8 10

Im(k

)

Re(k)

CorrectSpurious0

0

-2

-2

-4

-4

-6

-8

-8

-8

-10

-10

-10

To get axisymmetric resonances in corral model, compute:I Eigenvalues of a complex-scaled problemI Residuals in nonlinear eigenproblemI log10 A(k) A(k)

How do we know if we might miss something?

• A little complex analysis

If A nonsingular on , analytic inside, count eigs inside by

W(det(A)) =1

2i

ddz

ln det(A(z)) dz

= tr(

12i

A(z)1A(z) dz)

E = A A also analytic inside . By continuity,

W(det(A)) = W(det(A + E)) = W(det(A))

if A + sE nonsingular on for s [0,1].

• A general recipe

Analyticity of A and E +Matrix nonsingularity test for A + sE =Inclusion region for (A + E) +Eigenvalue counts for connected components of region

• Application: Matrix Rouch

A(z)1E(z) < 1 on = same eigenvalue count in

Proof:A(z)1E(z) < 1 = A(z) + sE(z) invertible for 0 s 1.

(Gohberg and Sigal proved a more general version in 1971.)

• Sensitivity and pseudospectra

-5

-4

-3

-2

-1

0

0 2 4 6 8 10

Im(k

)

Re(k)

CorrectSpurious0

0

-2

-2

-4

-4

-6

-8

-8

-8

-10

-10

-10

TheoremLet S = {z : A(z) A(z) < }. Any connected component of(Kext ,Mext ) strictly inside S contains the same number ofeigenvalues for A(k) and A(k).

Could almost certainly do better...

• For more