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On weak and strong solution operators for evolution equations coming from quadratic operators

Aleman, Alexandru LU and Viola, Joe LU (2018) In Journal of Spectral Theory 8(1). p.33-121
Abstract

We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplication-differentiation operators acting on L2.Rn/ which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behaviorwith the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the... (More)

We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplication-differentiation operators acting on L2.Rn/ which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behaviorwith the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Bargmann transform, Evolution equation, Fock space, Non-self-adjoint operator, Quadratic operator, Supersymmetric operator
in
Journal of Spectral Theory
volume
8
issue
1
pages
89 pages
publisher
European Mathematical Society Publishing House
external identifiers
  • scopus:85042782447
ISSN
1664-039X
DOI
10.4171/JST/191
language
English
LU publication?
yes
id
9faa70dd-febc-4bc1-936d-59474717c83e
date added to LUP
2018-03-15 15:27:57
date last changed
2019-02-20 11:10:48
@article{9faa70dd-febc-4bc1-936d-59474717c83e,
  abstract     = {<p>We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplication-differentiation operators acting on L<sup>2</sup>.R<sup>n</sup>/ which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behaviorwith the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.</p>},
  author       = {Aleman, Alexandru and Viola, Joe},
  issn         = {1664-039X},
  keyword      = {Bargmann transform,Evolution equation,Fock space,Non-self-adjoint operator,Quadratic operator,Supersymmetric operator},
  language     = {eng},
  number       = {1},
  pages        = {33--121},
  publisher    = {European Mathematical Society Publishing House},
  series       = {Journal of Spectral Theory},
  title        = {On weak and strong solution operators for evolution equations coming from quadratic operators},
  url          = {http://dx.doi.org/10.4171/JST/191},
  volume       = {8},
  year         = {2018},
}