scholarly journals Comment on “Exact constructions of square-root Helmholtz operator symbols: The focusing quadratic profile” [J. Math. Phys.41, 4881 (2000)]

2001 ◽  
Vol 42 (9) ◽  
pp. 4618-4623 ◽  
Author(s):  
P. M. Jordan
Keyword(s):  
Square Root ◽  
Helmholtz Operator ◽  
Operator Symbols ◽  
Math Phys

Exact constructions of square-root Helmholtz operator symbols: The focusing quadratic profile

2000 ◽  
Vol 41 (7) ◽  
pp. 4881-4938 ◽  
Author(s):  
Louis Fishman ◽  
Maarten V. de Hoop ◽  
Mattheus J. N. van Stralen
Keyword(s):  
Square Root ◽  
Helmholtz Operator ◽  
Operator Symbols

Uniform high-frequency approximations of the square root Helmholtz operator symbol

Wave Motion ◽  
1997 ◽  
Vol 26 (2) ◽  
pp. 127-161 ◽  
Author(s):  
Louis Fishman ◽  
A.K. Gautesen ◽  
Zhiming Sun
Keyword(s):  
High Frequency ◽  
Square Root ◽  
Helmholtz Operator ◽  
Operator Symbol

scholarly journals Eigenstate Thermalization Hypothesis for Wigner Matrices

2021 ◽  
Author(s):  
Giorgio Cipolloni ◽  
László Erdős ◽  
Dominik Schröder
Keyword(s):  
High Probability ◽  
Square Root ◽  
Commun Math Phys ◽  
Optimal Convergence Rate ◽  
Optimal Error ◽  
Wigner Matrix ◽  
Appl Math ◽  
Very High ◽  
Deterministic Matrix ◽  
Math Phys

AbstractWe prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020).

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