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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

    (Loading outs.mp4)

    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.

  • 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

  • Applicatio