Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

Belov, V. V.; Dobrokhotov, S. Yu.; Tudorovskii, T. Ya.

doi:10.1023/b:tamp.0000046563.43563.e6

Cited by 30 publications

(37 citation statements)

References 9 publications

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“…A detailed description of the solutions based on this algorithm and concrete physical results are not the goal of this work. These results present the contents of other publications (see, e.g., [84,85]). Here, in §5.3, we only very briefly describe the asymptotic solutions and the simplest physical results.…”

Section: Asymptotic Solutions To the Effective Equations Of Adiabaticsupporting

confidence: 70%

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Operator separation of variables for adiabatic problems in quantum and wave mechanics

2006

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We study linear problems of mathematical physics in which the adiabatic approximation is used in the wide sense. Using the idea that all these problems can be treated as problems with operator-valued symbol, we propose a general regular scheme of adiabatic approximation based on operator methods. This scheme is a generalization of the Born-Oppenheimer and Maslov methods, the Peierls substitution, etc. The approach proposed in this paper allows one to obtain "effective" reduced equations for a wide class of states inside terms (i.e., inside modes, subregions of dimensional quantization, etc.) with the possible degeneration taken into account. Next, applying the asymptotic methods in particular, the semiclassical approximation method, to the reduced equation, one can classify the states corresponding to a distinguished term (effective Hamiltonian). We show that the adiabatic effective Hamiltonian and the semiclassical Hamiltonian can be different, which results in the appearance of "nonstandard characteristics" while one passes to classical mechanics. This approach is used to construct solutions of several problems in wave and quantum mechanics, in particular, problems in molecular physics, solid-state physics, nanophysics, hydrodynamics.

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Section: Asymptotic Solutions To the Effective Equations Of Adiabaticsupporting

confidence: 70%

“…This discussion requires a special publication (see [85]) and probably not a single one. Here we only want very briefly to outline the structure of semiclassical asymptotics for some natural problems.…”

Section: Examples Of Asymptotic Solutions For Some Problems In Nanotubesmentioning

confidence: 99%

Operator separation of variables for adiabatic problems in quantum and wave mechanics

2006

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“…5. 2 We write the minus sign in the last term of this expression to correct the misprint in formula (1.18) in [4].…”

Section: Theorem Let Conditions 1-3 Be Satisfied Then the Functionmentioning

confidence: 98%

“…The Schrödinger-type equations with a potential independent of the electron position derived in [2] (also see [3]), just as any linear Schrödinger equations, have only localized propagating solutions inevitably spreading as they propagate through the entire waveguide length. The nonspreading of localized solutions (the ballistic transport in strongly extensive waveguides) can thus occur only if the nonlinear effects are taken into account.…”

Section: Introductionmentioning

confidence: 99%

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Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

et al. 2008

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We consider the nonlinear Schrödinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrödinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.

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Quantum scattering by a Viviani’s curve

Maioli

Schmidt

Azado

2022

Z. Angew. Math. Phys.

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Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

Cited by 30 publications

References 9 publications

Operator separation of variables for adiabatic problems in quantum and wave mechanics

Operator separation of variables for adiabatic problems in quantum and wave mechanics

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

Quantum scattering by a Viviani’s curve

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