TY - JOUR

T1 - Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall

AU - Duca, Alessandro

AU - Joly, Romain

AU - Turaev, Dmitry

N1 - Funding Information:
This work has been initiated by a discussion in the fruitful atmosphere of the Hale conference in Higashihiroshima. This work was supported by the RSF (Grant Nos. 19-11-00280 and 19-71-10048) and the Project ISDEEC (ANR-16-CE40-0013). D.T. acknowledges support from the EPSRC and the Russian Ministry of Science and Education (Project No. 2019-220-07-4321).
Publisher Copyright:
© 2020 Author(s).

PY - 2020/10/1

Y1 - 2020/10/1

N2 - We study the Schrödinger equation itψ = -Δψ + Vψ on L2((0,1),C) where V is a very high and localized potential wall. We consider the process where the position and the height of the wall change as follows: First, the potential increases from zero to a very large value, and so a narrow potential wall is formed and almost splits the interval into two parts; then, the wall moves to a different position, after which the height of the wall decreases to zero again. We show that even though the rate of variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the instantaneous energy eigenstates. Furthermore, we consider potentials with several narrow walls and show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schrödinger equation by means of a soft, quasi-adiabatic variation of the potential.

AB - We study the Schrödinger equation itψ = -Δψ + Vψ on L2((0,1),C) where V is a very high and localized potential wall. We consider the process where the position and the height of the wall change as follows: First, the potential increases from zero to a very large value, and so a narrow potential wall is formed and almost splits the interval into two parts; then, the wall moves to a different position, after which the height of the wall decreases to zero again. We show that even though the rate of variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the instantaneous energy eigenstates. Furthermore, we consider potentials with several narrow walls and show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schrödinger equation by means of a soft, quasi-adiabatic variation of the potential.

UR - http://www.scopus.com/inward/record.url?scp=85094612608&partnerID=8YFLogxK

U2 - 10.1063/5.0005399

DO - 10.1063/5.0005399

M3 - Article

AN - SCOPUS:85094612608

VL - 61

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

M1 - 0005399

ER -