arXiv subjects

Andrey V. Sokolov

Publications and source records attributed to Andrey V. Sokolov.

8 recordsLinked to original sources

arXiv2023

A simple catch: thermal fluctuations enable hydrodynamic trapping of microrollers by obstacles

It is known that obstacles can hydrodynamically trap bacteria and synthetic microswimmers in orbits, where the trapping time heavily depends on the swimmer flow field and noise is needed to escape the trap. Here, we use experiments and simulations to investigate the trapping of microrollers by obstacles. Microrollers are rotating particles close to a bottom surface, which have a prescribed propulsion direction imposed by an external rotating magnetic field. The flow field that drives their motion is quite different from previously studied swimmers. We found that the trapping time can be controlled by modifying the obstacle size or the colloid-obstacle repulsive potential. We detail the mechanisms of the trapping and find two remarkable features: The microroller is confined in the wake of the obstacle, and it can only enter the trap with Brownian motion. While noise is usually needed to escape traps in dynamical systems, here, we show that it is the only means to reach the hydrodynamic attractor.

cond-mat.soft↗

arXiv2018

Polynomial supersymmetry for matrix Hamiltonians: proofs

In this paper we continue studying of matrix $n\times n$ linear differential intertwining operators. The problems of minimization and of reducibility of matrix intertwining operators are considered and criterions of weak minimizability and of regular reducibility are proven. It is shown that in contrast to the scalar case $n=1$ there are for any $n\geqslant2$ regularly absolutely irreducible matrix intertwining operators of any order $N\geqslant2$, {\it i.e.} operators which cannot be factorized into a product of a matrix intertwining operators of lower orders even with a pole singularity(-ies) into coefficients. The theorem is proven on existence for any matrix intertwining operator $Q_N^-$ of the order $N$ a matrix differential operator $Q_{N'}^+$ of other, in general, order $N'$ that intertwines the same Hamiltonians $H_+$ and $H_-$ as $Q_N^-$ in the opposite direction and such that the products $Q_{N'}^+Q_N^-$ and $Q_N^-Q_{N'}^+$ are identical polynomials of $H_+$ and $H_-$ respectively. Polynomial algebra of supersymmetry corresponding to this case is constructed.

math-ph↗

arXiv2014

Minimal Realizations of Supersymmetry for Matrix Hamiltonians

The notions of weak and strong minimizability of a matrix intertwining operator are introduced. Criterion of strong minimizability of a matrix intertwining operator is revealed. Criterion and sufficient condition of existence of a constant symmetry matrix for a matrix Hamiltonian are presented. A method of constructing of a matrix Hamiltonian with a given constant symmetry matrix in terms of a set of arbitrary scalar functions and eigen- and associated vectors of this matrix is offered. Examples of constructing of $2\times2$ matrix Hamiltonians with given symmetry matrices for the cases of different structure of Jordan form of these matrices are elucidated.

math-ph↗

arXiv2014

Spectral Design for Matrix Hamiltonians: Different Methods of Constructing of a Matrix Intertwining Operator

We study intertwining relations for $n\times n$ matrix non-Hermitian, in general, one-dimensional Hamiltonians by $n\times n$ matrix linear differential operators with nondegenerate coefficients at $d/dx$ in the highest degree. Some methods of constructing of $n\times n$ matrix intertwining operator of the first order of general form are proposed and their interrelation is examined. As example we construct $2\times2$ matrix Hamiltonian of general form intertwined by operator of the first order with the Hamiltonian with zero matrix potential. It is shown that one can add for the final $2\times2$ matrix Hamiltonian with respect to the initial matrix Hamiltonian with the help of intertwining operator of the first order either up to two bound states for different energy values or up to two bound states described by vector-eigenfunctions for the same energy value or up to two bound states described by vector-eigenfunction and associated vector-function for the same energy value.

math-ph↗

arXiv2013

Polynomial Supersymmetry for Matrix Hamiltonians

We study intertwining relations for matrix one-dimensional, in general, non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any matrix intertwining operator Q_N^- of minimal order N there is a matrix operator Q_{N'}^+ of different, in general, order N' that intertwines the same Hamiltonians as Q_N^- in the opposite direction and such that the products Q_{N'}^+Q_N^- and Q_N^-Q_{N'}^+ are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and of reducibility of a matrix intertwining operator are considered and the criteria of minimizability and of reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.

quant-ph↗

arXiv2011

Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum

Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined.

math-ph↗

arXiv2011

Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs

This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.

math-ph↗

arXiv2009

Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics

When several inequivalent supercharges form a closed superalgebra in Quantum Mechanics it entails the appearance of hidden symmetries of a Super-Hamiltonian. We examine this problem in one-dimensional QM for the case of periodic potentials and potentials with finite number of bound states. After the survey of the results existing in the subject the algebraic and analytic properties of hidden-symmetry differential operators are rigorously elaborated in the Theorems and illuminated by several examples.

hep-th↗