Alexander Zhalij
In this paper we construct integrable three-dimensional quantum-mechanical systems with magnetic fields, admitting pairs of commuting second-order integrals of motion. The case of Cartesian coordinates is considered. Most of the systems obtained are new and not related to the separation of variables in the corresponding Schrödinger equation.
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We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems sepa…
Integrable systems in magnetic fields: the generalized parabolic cylindrical case
本文研究带向量势的二次可积系统分类，在广义抛物柱情形中发现3个新系统，并通过参数与零磁场极限验证非子群型积分仅能平凡关联可分离系统的猜想。
The goal of this thesis is the search for integrable and superintegrable systems with magnetic field. We formulate the quantum mechanical determining equations for second order integrals of motion in the cylindrical coordinates and we find all quadratically integrable systems of the cylindrical type. Among them we search for systems admitting additional integrals of motion. We find all systems with an additional first order integral both in classical and quantum mechanics. It turns out that all these systems have already been known and no other exist. We also find all systems with an additiona…
Two integrable cases of two-dimensional Schrödinger equation with a magnetic field are proposed. Using the polar coordinates and the symmetrical gauge, we will obtain solutions of these equation through Biconfluent and Confluent Heun functions. The quantization rules will be derived for both systems under consideration.
We consider magnetic geodesic flows on the 2-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a Semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.
We present a new family of integrable versions of the Euler two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole of arbitrary charge. The new systems have very special algebraic potential and additional integral quadratic in momenta, both in classical and quantum versions.
We consider a class of Hamiltonian systems in 3 degrees of freedom, with a particular type of quadratic integral and which includes the rational Calogero-Moser system as a particular case. For the general class, we introduce separation coordinates to find the general separable (and therefore Liouville integrable) system, with two quadratic integrals. This gives a coupling of the Calogero-Moser system with a large class of potentials, generalising the series of potentials which are separable in parabolic coordinates. Particular cases are {\em superintegrable}, including Kepler and a resonant os…
We classify certain integrable (both classical and quantum) generalisations of Dirac magnetic monopole on topological sphere $S^2$ with constant magnetic field, completing the previous local results by Ferapontov, Sayles and Veselov. We show that there are two integrable families of such generalisations with integrals, which are quadratic in momenta. The first family corresponds to the classical Clebsch systems, which can be interpreted as Dirac magnetic monopole in harmonic electric field. The second family is new and can be written in terms of elliptic functions on sphere $S^2$ with very spe…
Integrable quantum mechanical systems with magnetic fields are constructed in two-dimensional Euclidean space. The integral of motion is assumed to be a first or second order Hermitian operator. Contrary to the case of purely scalar potentials, quadratic integrability does not imply separation of variables in the Schroedinger equation. Moreover, quantum and classical integrable systems do not necessarily coincide: the Hamiltonian can depend on the Planck constant in a nontrivial manner.
Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields
本文构造了磁场中积分具有特定领先阶结构的所有非标准可积系统，发现它们可表示为一个多参数系统族，并在参数特定极限下与已知标准系统相交，从而成为最小超可积系统。
Right-invariant geodesic flows on manifolds of Lie groups associated with 2-cocycles of corresponding Lie algebras are discussed. Algebra of integrals of motion for magnetic geodesic flows is considered and necessary and sufficient condition of integrability in quadratures is formulated. Canonic forms for 2-cocycles of all 4-dimensional Lie algebras are given and integrable cases among them are separated.
New classes of quadratically integrable systems in magnetic fields: the generalized cylindrical and spherical cases
本文研究三维磁场中具有二次运动积分的可积与超可积系统，推广柱面型和球面型积分，发现三个柱面可积系统及一个不可正交分离的最小超可积系统。
The factorisation method for Schrödinger operators with magnetic fields on a two-dimensional surface $M^2$ with non-trivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as Dirac magnetic monopole and Landau problem. The global geometric aspects and related spectral properties of the operators from the factorisation chains are discussed in details. We also consider the Laplace transformations on a curved surface and extend the class of Schrödinger operators with two integrable levels introduced in the f…
Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry
本文研究三维黎曼空间中磁场内经典哈密顿系统的几何，证明其具有辛-哈安特耶斯结构，并利用该结构算法求解分离变量及构造新的可积模型。
We establish the most general class of spin-1/2 integrable Richardson-Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin-1/2 relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Richardson-Gaudin models. The full set of conserved charges are constructed explicitly and shown to satisfy a set of quadratic equations, allowing for the numerical treatment of a fully anisotropic central spin in an external magnetic field. While this…
The aim of the present article is to construct quadratically integrable three dimensional systems in non-vanishing magnetic fields which possess so-called non-subgroup type integrals. The presence of such integrals means that the system possesses a pair of integrals of motion in involution which are (at most) quadratic in momenta and whose leading order terms, that are necessarily elements of the enveloping algebra of the Euclidean algebra, are not quadratic Casimir operators of a chain of its subalgebras. By imposing in addition that one of the integrals has the leading order term $L_z^2$ we …
We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space $\mathbb{E}_3$ with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form $H =\frac{1}{2}\left(\vec{p}+\vec{A}(\vec{x})\right)^2+W(\vec{x})$, $X_1=(p_ϕ^A)^2+s_1^r(r, ϕ, Z)p_r^A+s_1^ϕ(r, ϕ, Z)p_ϕ^A+s_1^Z(r, ϕ, Z)p_Z^A+m_1(r,ϕ,Z)$, $X_2=(p_Z^A)^2+s_2^r(r, ϕ, Z)p_r^A+s_2^ϕ(r, ϕ, Z)p_ϕ^A+s_2^Z(r, ϕ, Z)p_Z^A+m_2(r,ϕ,Z)$. Infinite families of such systems are found, in general depending on arbitrary functions or parameters. This leaves open the possibil…
The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong-Wong type. Under certain conditions these generalizations remain integrable. We further set up a gauge theory with a product gauge group, two complex scalar fields and a general charge matrix. The second species of vortices, when frozen, are interpreted as the magnetic impurity for all five vortex equations. We then give a geometric compatibility condition, which enables us to remove the constant term in all the equations. This is similar to the reduction from the Taubes…
We consider a four dimensional field theory with target space being CP^N which constitutes a generalization of the usual Skyrme-Faddeev model defined on CP^1. We show that it possesses an integrable sector presenting an infinite number of local conservation laws, which are associated to the hidden symmetries of the zero curvature representation of the theory in loop space. We construct an infinite class of exact solutions for that integrable submodel where the fields are meromorphic functions of the combinations (x^1+i x^2) and (x^3+x^0) of the Cartesian coordinates of four dimensional Minkows…
We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and over a sphere in $\mathbb R^n$ as examples of gyroscopic $SO(n)$--Chaplygin systems. We describe…
The problem of $N$ particles interacting through pairwise central forces is notoriously intractable for $N\geq3$. Some quite remarkable specific cases have been solved in one dimension, whereas higher-dimensional exactly solved systems involve velocity-dependent or many-body forces. Here we show that the guiding center approximation---valid for charges moving in two dimensions in a strong constant magnetic field---simplifies the three-body problem for an arbitrary interparticle interaction invariant under rotations and translations and makes it solvable by quadratures. This includes a broad va…
We have found a family of solvable nineteen vertex model with statistical configurations invariant by the time reversal symmetry within a systematic study of the respective Yang-Baxter relation. The Boltzmann weights sit on a degree seven algebraic threefold which is shown birationally equivalent to the three-dimensional projective space. This permits to write parameterized expressions for both the transition operator and the $\mathrm{R}$-matrix depending on three independent affine spectral parameters. The Hamiltonian limit tells us that the azimuthal magnetic field term is connected with the…
In this paper we give a short survey of recent results on algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov.
We consider the problem of motion of a rigid body about a fixed point under the action of an axisymmetric combination of potential and gyroscopic forces. We introduce a new integrable case, valid on zero level of the cyclic integral, that generalizes the known case of motion of a body in liquid due to Chaplygin and its subsequent generalization by Yehia. Apart from certain singular potential terms, the new case involves finite potential and gyroscopic forces, which admit physical interpretation as resulting from interaction of mass, magnetized parts and electric charges on the body with gravit…
We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method we show that it behaves well with respect to the infinite tensor products. This construction subsumes examples coming from the study of magnetic Weyl calculus, the magnetic pseudo-differential Weyl calculus, the metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of so…
Integrability, Normal Forms, and Magnetic Axis Coordinates
本文将磁场视为预辛系统，证明可积磁场在椭圆与双曲磁轴附近存在精确光滑正规形，并构造近轴 Hamada 和 Boozer 坐标。
We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for "nearly integrable" magnetic fields up to a "critical" size. In…
We describe a class of completely integrable $G$-invariant magnetic geodesic flows on (co)adjoint orbits of a compact connected Lie group $G$ with magnetic field given by the Kirillov-Konstant 2-form.
The equilibrium equations for the isotropic Kirchhoff rod are known to form an integrable system. It is also known that the effects of extensibility and shearability of the rod do not break the integrable structure. Nor, as we have shown in a previous paper does the effect of a magnetic field on a conducting rod. Here we show, by means of Mel'nikov analysis, that, remarkably, the combined effects do destroy integrability; that is, the governing equations for an extensible current-carrying rod in a uniform magnetic field are nonintegrable. This result has implications for possible configuration…
Using the well known position-dependent mass (PDM) von Roos Hamiltonian, Dutra and Oliveira (2009 J. Phys. A: Math. Theor. 42 025304) have studied the problem of two-dimensional PDM particles in the presence of magnetic fields. They have reported exact solutions for the wavefunctions and energies. In the first part of their study "PDM-Shrödinger equation in two-dimensional Cartesian coordinates", they have used the so called Zhu and Kroemer's ordering α=-1/2=γ and \b{eta}=0 <cite>5</cite>. While their treatment for this part is correct beyond doubt, their treatment of second part "PDM in a mag…
We calculate the condensate and the vacuum current density induced by external static magnetic fields in (2+1)-dimensions. At the perturbative level, we consider an exponentially decaying magnetic field along one cartesian coordinate. Non-perturbatively, we obtain the fermion propagator in the presence of a uniform magnetic field by solving the Schwinger-Dyson equation in the rainbow-ladder approximation. In the large flux limit, we observe that both these quantities, either perturbative (inhomogeneous) and non-perturbative (homogeneous), are proportional to the external field, in agreement wi…
An integrable Anderson-impurity problem embedded in the one-dimensional Hubbard model
本文在量子反散射框架下构造了边界嵌入单个Anderson杂质的可积一维Hubbard模型，用嵌套Bethe ansatz精确求解，识别边界束缚态、确定基态相图，并揭示稀电子极限下近自由局域磁矩与有限密度下近藤屏蔽行为。
The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper we explicitly calculate its time evolution and the corresponding action-angle variables. This calculation is facilitated by splitting the six degrees of freedom into three internal and three external variables, such that the internal variables evolve autonomously. Their oscillations can be explicitly calculated in terms of the Weierstrass elliptic function. We test our results by means of an example and comparison with direct numerical integr…
We consider the equilibrium equations for a conducting elastic rod placed in a uniform magnetic field, motivated by the problem of electrodynamic space tethers. When expressed in body coordinates the equations are found to sit in a hierarchy of non-canonical Hamiltonian systems involving an increasing number of vector fields. These systems, which include the classical Euler and Kirchhoff rods, are shown to be completely integrable in the case of a transversely isotropic rod; they are in fact generated by a Lax pair. For the magnetic rod this gives a physical interpretation to a previously prop…
Integrable Spherical Brane Model at Large $N$
研究一个最简单的具有边界的二维可积量子场论：bulk中N个非紧缩标量，被非线性约束在半径为1/√g的(N-1)球面上。大N情况下对应高自旋杂质或多通道情形，添加边界磁场保持可积性并验证了Lukyanov和Zamolodchikov的猜想。
Instantons, renormalons and the theta angle in integrable sigma models
本研究针对在$\vartheta=0$和$\vartheta=\pi$下可积的含theta角sigma模型，通过Bethe ansatz计算磁场下基态能量的非微扰修正，分析theta角的影响，给出弱耦合下带磁场$O(3)$ sigma模型拓扑磁化率的显式非微扰公式。
In this paper, we study the surface energy of the one-dimensional supersymmetric $t-J$ model with unparallel boundary magnetic fields, which is a typical $U(1)$-symmetry broken quantum integrable strongly correlated electron system. It is shown that at the ground state, the contribution of inhomogeneous term in the Bethe ansatz solution of eigenvalues of transfer matrix satisfies the finite size scaling law $L^β$ where $β<0$. Based on it, the physical quantities of the system in the thermodynamic limit are calculated. We obtain the patterns of Bethe roots and the analytical expressions of dens…
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