Resolvent estimates for elliptic quadratic differential operators

Hitrik, Michael; Sjöstrand, Johannes; Viola, Joe

doi:10.2140/apde.2013.6.181

Cited by 25 publications

(58 citation statements)

References 17 publications

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“…Real-side quadratic operators. Much of the following discussion can be found in previous works including [26], [15], [17], and [32]. Let q(x, ξ) : R 2n → C be a quadratic form.…”

Section: Real-side Equivalencementioning

confidence: 99%

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

Aleman

Viola

2018

J. Spectr. Theory

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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 multiplicationdifferentiation operators acting on L 2 (R n ) 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 behavior with 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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“…Real-side quadratic operators. Much of the following discussion can be found in previous works including [26], [15], [17], and [32]. Let q(x, ξ) : R 2n → C be a quadratic form.…”

Section: Real-side Equivalencementioning

confidence: 99%

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

Aleman

Viola

2018

J. Spectr. Theory

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“…First of all, we always assume ω = 0 in order to define H as a perturbation of the harmonic oscillator H sa defined in (32); without loss of generality, let us take ω > 0. Second, we need to impose a condition on the smallness of α and β in order to ensure that the extra unbounded terms r := α a 2 + β (a * ) 2 added in (41) to ω a * a = ωH sa − ω do not completely change the character of the operator H sa ; see Section 7.8 for a discussion of the relevant condition in a more general setting. Expressing annihilation and creation operators in terms of x and d/dx, we obtain an equivalent form…”

Section: The Gauged Oscillator (Swanson's Model)mentioning

confidence: 99%

“…To find the eigenvalues of H, we observe that H is formally similar to a self-adjoint harmonic oscillator. Indeed, substituting (34) to (41) and completing the square, we find…”

Section: The Gauged Oscillator (Swanson's Model)mentioning

confidence: 99%

Pseudospectra in non-Hermitian quantum mechanics

Krejčiřı́k

Siegl

Tater

et al. 2015

Journal of Mathematical Physics

120

124

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We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.Comment: version accepted for publication in J. Math. Phys.: criterion excluding basis property (Proposition 6) added, unbounded time-evolution discussed, new reference

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“…So far, non-Hermitian harmonic systems have been mostly analysed from the spectral point of view or in the context of PT symmetry, see for example [Sjö74,§3], [Dav99b] or more recently [CGHS12,KSTV15]. It has been proven that the condition number of the eigenvalues of non-Hermitian harmonic systems grows rapidly with respect to their size [DK04,Hen14], while spectral asymptotics have been obtained for skew-symmetric perturbations of harmonic oscillators as well as for non-selfadjoint system with double characteristics [GGN09, HP13,HSV13]. The semigroup of non-selfadjoint quadratic operators has been analysed in [Pra08] and [AV15,Vio16].…”

Section: Introductionmentioning

confidence: 99%

Non-Hermitian propagation of Hagedorn wavepackets

Lasser

Schubert

Troppmann

2018

Journal of Mathematical Physics

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We investigate the time evolution of Hagedorn wavepackets by non-Hermitian quadratic Hamiltonians. We state a direct connection between coherent states and Lagrangian frames. For the time evolution a multivariate polynomial recursion is derived that describes the activation of lower lying excited states, a phenomenon unprecedented for Hermitian propagation. Finally we apply the propagation of excited states to the Davies-Swanson oscillator.Re(ZZ * )ΩZ = iZ , Re(ZZ * )ΩZ = −iZ , so that (Re(ZZ * )Ω) 2 = −Id 2n .Proof. Writing π L + πL = Id 2n in terms of Z, we obtain − Im(ZZ * )Ω T = Id 2n . Hence, Im(ZZ * ) = Ω T . This implies symplecticity of the real part, sinceChecking positive definiteness, we see z · Re(ZZ * )z = 1 2 z · (ZZ * z +ZZ T z) = |Z * z| ≥ 0

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Resolvent estimates for elliptic quadratic differential operators

Abstract: Sharp resolvent bounds for non-selfadjoint semiclassical elliptic quadratic differential operators are established, in the interior of the range of the associated quadratic symbol.

Cited by 25 publications

References 17 publications

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

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

Pseudospectra in non-Hermitian quantum mechanics

Non-Hermitian propagation of Hagedorn wavepackets

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