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Jan. 21th, 2024
研究集会「無限粒子系、確率場の諸問題XVIII」
Department of Physics, Chuo University, Tokyo
Saori MORIMOTO Joint work with M.Katori and T.Shirai
Eigenvalue and pseudospectrum processes generated nonnormal Toeplitz matrices with perturbation
arXiv/math-ph/2401.08129
Outline
0. Introduction
1. Shift dynamics with uniform perturbation (Model 1)
2. Toeplitz matrices with uniform perturbation (Model 2)
3. Future problems
4. References
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0. Introduction
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0.1. The matrix-valued BM starts from the null matrix
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0.2. The matrix-valued BM starts from the shift matrix
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0.3. Shift matrix with Gaussian random perturbation
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0.4. Shift dynamics with Gaussian random perturbation
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Appearance of devil’s staircase
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Appearance of devil’s staircase
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1. Shift dynamics with uniform perturbation (Model 1)
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1.1. Theorems on the eigenvalue processes
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Theorem 1.1
・・・ (１)
Solution of the equation (1)
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Theorem 1.1
・・・ (１)
1.1. Theorems on the eigenvalue processes
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Theorem 1.1
・・・ (１)
1.1. Theorems on the eigenvalue processes
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Theorem 1.1
・・・ (１)
1.1. Theorems on the eigenvalue processes
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Theorem 1.1
・・・ (１)
1.2. Definition of pseudospectrum
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Example of the resolvent norm from [3]
1.3. Eigenvalues and pseudospectrum
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1.3. Eigenvalues and pseudospectrum
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1.3. Eigenvalues and pseudospectrum
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1.3. Eigenvalues and pseudospectrum
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1.3. Eigenvalues and pseudospectrum
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1.4. Conjecture 1
Numerical calculation suggests eigenvalue-like (peak) structures in the pseudospectrum. Analytic study is desired.
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eigenvalues
pseudospectrum
eigenvalue-like (peak) structures
2. Toeplitz matrices with uniform perturbation (Model 2)
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Compare Model 1 and Model 2
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Model 2
Model 1
Compare Model 1 and Model 2
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Model 2
Model 1
2.1. Theorems on the eigenvalue processes
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Theorem 2.1
・・・ (2)
Compare Model 1 and Model 2
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Theorem 1.1
・・・ (１)
Theorem 2.1
・・・ (２)
Solution of the equation (2)
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Theorem 2.1
・・・ (２)
2.1. Theorems on the eigenvalue processes
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Theorem 2.1
・・・ (２)
2.1. Theorems on the eigenvalue processes
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Theorem 2.1
・・・ (２)
2.1. Theorems on the eigenvalue processes
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Theorem 2.1
・・・ (２)
2.2. Symbol curves of Toeplitz operators
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2.2. Symbol curves of Toeplitz operators
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2.2. Symbol curves of Toeplitz operators
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2.2. Symbol curves of Toeplitz operators
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Theorems for banded Toeplitz matrices
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Lloyd N. Trefethen, Mark Embree, `Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators', Princeton University Press, Princeton and Oxford, 2005
Proposition [ Trefethen-Embree 2005 ]
Theorem [ Trefethen-Embree 2005 ]
2.3. Symbol curve of Model 2
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Symbol curves
Compare with the inner structure and symbol curve
2.4. Conjecture 2
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inner part of the “eigenvalues”
outermost closed simple curve
“eigenvalues”
inner part of symbol curve
exact eigenvalues
2.5. Ongoing work
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inner part of the “eigenvalues”
outermost closed simple curve
“eigenvalues”
inner part of symbol curve
exact eigenvalues
3. Future problems
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3. Future problems
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4. References
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Thank you very much for your attention.
[1] Burda, Z., Grela, J., Nowak, M. A., Tarnowski, W., Warchol, P.: Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics. Nucl. Phys. B 897, 421-447 (2015)
[2] Trefethen, L. N., Embree, M.: Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators, vol. 1. Princeton University Press, Princeton (2005)
[3] Lloyd N. Trefethen, Mark Embree, `Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators', Princeton University Press, Princeton and Oxford, 2005
arXiv/math-ph/2401.08129