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Resonances:Interpretation, Computation, and Perturbation
David Bindel
Department of Computer ScienceCornell University
14 March 2011
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The quantum corral
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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.
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Electrons unbound
For a finite barrier, electrons can escape!Not a bound state (conventional eigenmode).
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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
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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.
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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.
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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)
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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)
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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
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Resonances and transients
(Loading outs.mp4)
Lavf52.31.0
outs.mp4Media File (video/mp4)
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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
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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.
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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.
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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.
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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)
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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/
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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.
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Is it that easy?
400 200 0 200 40040
30
20
10
0
10
20
All eigenvaluesChecked eigenvalues
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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
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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
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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.
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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
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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).
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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 .
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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?
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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 ).
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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)
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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}
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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
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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?
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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].
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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
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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.)
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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...
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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/