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

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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.

Computational tradeoffs

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

More information at

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

I Links to tutorial notes on resonances with Maciej ZworskiI Matscat code for computing resonances for 1D problemsI These slides!

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