A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

Lungenstrass, Tomas; Raikov, Georgi

doi:10.1007/s00023-013-0285-1

Cited by 6 publications

(8 citation statements)

References 24 publications

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“…We can also mention papers [19,26], where the authors investigated properties of eigenfunctions of perturbed Hamiltonians, and in [20,23,24,[30][31][32]37] asymptotics of the eigenvalues for perturbed Landau Hamiltonians were described.…”

Section: Introductionmentioning

confidence: 99%

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Ruzhansky

Tokmagambetov

2016

Lett Math Phys

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In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with time-dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional-type solutions under conditions when such solutions also exist.

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Section: Introductionmentioning

confidence: 99%

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Ruzhansky

Tokmagambetov

2016

Lett Math Phys

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“…Now, let us show that 40) for some c, and for all t ∈ [0, T ] and ε ∈ (0, 1]. By the simple calculations we get…”

Section: Case I4: Proof Of Theorem 24 (A)mentioning

confidence: 93%

“…Their perturbations have been investigated in [35,44], and the asymptotic behaviour of the eigenvalues was analysed in [37,40,41,[48][49][50]57]. The results of this paper apply for the Cauchy problem (1.1) for the operator L from (3.2).…”

Section: Landau Hamiltonian In 2dmentioning

confidence: 99%

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

Ruzhansky

Tokmagambetov

2017

Arch Rational Mech Anal

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Given a Hilbert space H, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on H. We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of "very weak solutions" to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on R n , uniformly elliptic operators of different orders on domains, Hörmander's sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

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“…Remark: The Berezin-Toeplitz operators related to the Fock-Segal-Bargmann holomorphic subspace of L 2 (R 2 ), and their generalizations corresponding to higher Landau levels, are known to play an important role in the spectral and scattering theory of quantum Hamiltonians in constant magnetic fields (see e.g. [31,32,20,13,30,14,29]…”

Section: Berezin Theory's Point Of Viewmentioning

confidence: 99%

Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian

Bruneau

Raikov

2018

ASY

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We consider harmonic Toeplitz operators T V = P V : H(Ω) → H(Ω) where P : L 2 (Ω) → H(Ω) is the orthogonal projection onto H(Ω) = u ∈ L 2 (Ω) | ∆u = 0 in Ω , Ω ⊂ R d , d ≥ 2, is a bounded domain with boundary ∂Ω ∈ C ∞ , and V : Ω → C is an appropriate multiplier. First, we complement the known criteria which guarantee that T V is in the pth Schatten-von Neumann class S p , by simple sufficient conditions which imply T V ∈ S p,w , the weak counterpart of S p . Next, we consider symbols V ≥ 0 which have a regular power-like decay of rate γ > 0 at ∂Ω, and we show that T V is unitarily equivalent to a pseudo-differential operator of order −γ, self-adjoint in L 2 (∂Ω). Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for T V , and establish a sharp remainder estimate. Further, we assume that Ω is the unit ball in R d , and V = V is compactly supported in Ω, and investigate the eigenvalue asymptotics of the Toeplitz operator T V . Finally, we introduce the Krein Laplacian K, self-adjoint in L 2 (Ω), perturb it by a multiplier V ∈ C(Ω; R), and show that σ ess (K + V ) = V (∂Ω). Assuming that V ≥ 0 and V |∂Ω = 0, we study the asymptotic distribution of the discrete spectrum of K ± V near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator T V .

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A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

Cited by 6 publications

References 24 publications

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian

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