On a class of anharmonic oscillators

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous leftinvariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.

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“…Other examples are a large class of harmonic and anharmonic oscillators in all dimension, see e.g. [CDR18].…”

Section: Examplesmentioning

confidence: 99%

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Ruzhansky

Tokmagambetov

Torebek

2019

Journal of Inverse and Ill-Posed Problems

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“…The operator π λ,µ (A) has discrete spectrum (see e.g. [8]), and if N A (s) denotes the number of its eigenvalues not exceeding s, then…”

Section: The Engel Groupmentioning

confidence: 99%

“…Let us note that in [1, Example 7.5] the proof of the analogous results to those in this note, in the setting of the Heisenberg group H n , relies on the fact that the (global) symbol of the canonical sub-Laplacian on H n , is the harmonic oscillator, and thus the eigenfunctions (the well-known Hermite functions), as well as the corresponding eigenvalues are known explicitly. However, in the examples of this paper we proceed without knowing the exact eigenvalues of the symbol of the Fourier multiplier in each setting using estimates for the asymptotic behaviour of the eigenvalue counting function N(s) for anharmonic oscillators − d 2 du 2 + V (u) on L 2 (R), using some analysis of [8].…”

Section: Introductionmentioning

confidence: 99%

A note on spectral multipliers on Engel and Cartan groups

Chatzakou

2022

Proc. Amer. Math. Soc.

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The aim of this short note is to give examples of L p L^p - L q L^q bounded spectral multipliers for operators involving left-invariant vector fields and their inverses, in the settings of Engel and Cartan groups. The interest in such examples lies in the fact that a group does not have to have flat co-adjoint orbits, and that the considered operator is not related to the usual sub-Laplacian. The discussed examples show how one can still obtain L p L^p - L q L^q estimates for similar operators in such settings. As immediate consequences, one gets the corresponding Sobolev-type inequalities and heat kernel estimates.

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“…Observe that π λ,µ (L B 4 ) is a particular case of the anharmonic oscillator considered in [CDR18].…”

Section: Preliminaries On the Groupsmentioning

confidence: 99%

“…is a particular case of the anharmonic oscillator on S(R) ⊂ L 2 (R). Thus, it is known, see [CDR18], that its spectrum is discrete, and if N A (s) denotes the number of its eigenvalues that are less that s, then…”

Section: Preliminaries On the Groupsmentioning

confidence: 99%

Quantizations on the Engel and the Cartan groups

Chatzakou¹

2020

Preprint

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This work aims to develop a global quantization in the concrete settings of two graded nilpotent Lie groups of 3-step; namely of the Engel group and the Cartan group. We provide a preliminary analysis on the structure and the representations of the aforementioned groups, and their corresponding Lie algebras. In addition, the explicit formulas for the difference operators in the two settings are derived, constituting the necessary prerequisites for the constructions of the Ψ m ρ,δ classes of symbols in both cases. In the case of the Engel group, the relation between the Kohn-Nirenberg quantization and the representations of the Engel group enables us to express operators in this setting in terms of quantization of symbols in the Euclidean space. We illustrate the L p − L q boundedness for spectral multipliers theorems with particular examples of operators in both groups, that yield appropriate Sobolev-type inequalities. As a further application of the above, we get some consequences for the L p − L q norm for the heat kernel of those particular examples of operators in both settings. Contents 16 4.1. The case of the Engel group 17 4.2. The case of the Cartan group 20 5. L p − L q mutlipliers on the Engel and the Cartan groups 26 Appendix A. Plancherel measure on the dual of the Engel group 32 References 34

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On a class of anharmonic oscillators

Cited by 17 publications

References 28 publications

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

A note on spectral multipliers on Engel and Cartan groups

Quantizations on the Engel and the Cartan groups

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