Eigenfunction expansions of operators admitting separation of an infinite number of variables

Let A_k be a self-adjoint operator in a Hilbert space H_k (k = 1, 2, …) and let $\text{L}$ be an operator of the form $\text{L}\text{= A}{}_{\mathrm{r}}\text{? 1 ? 1 ? … + 1 ? A}{}_2\text{? 1 ? 1 ? … + …}$ acting in the infinite tensor product $\text{?}\text{k=1}\text{∞}\text{H}{}_{\mathrm{k}}$. We construct the spectral theory of these operators. In particular, the expansion is generalized eigenvectors of this operator is constructed using the eigenvectors of the operators A_k.     

Commutative subalgebras of three first-order symmetry operators and separation of variables in the wave equation

The problem of complex separation of variables in the wave equation is considered in four-dimensional Minkowskii space-time. In contrast to the known series of researches by Kalnins and Miller (see Ref. Zh., Fiz., 2B9 (1978); 1B208 and 1B209 (1979), e.g.), underlying this research is a theorem on the necessary and sufficient conditions of total separation of variables in the non-parabolic V. N. Shapovalov equation (Differents. Uravn.,16, No. 10, 1864–1874 (1980)). Nonequivalent complete sets of three differential first-order symmetry operators are constructed, appropriate coordinate systems are found, and complete separation of variables is performed in the wave equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 79–84, May, 1990.     

Complete sets of symmetry operators containing a second-order operator and the problem of separation of variables in the wave equation

For the wave equation in Minkowski space, a space is defined of nontrivial local second-order differential symmetry operators. The algebraic conditions, which, in accordance with the general theorems on the separation of variables, must be satisfied by the commutative subalgebras, including two first-order operators and second-order operator, are formulated in coordinate-free form. On the basis of these subalgebras, there are obtained all the complete sets of symmetry operators of types (2.0), (2.1). Sets are presented which do not have analogues in papers of other authors.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 115–119, April, 1991.     

Two-dimensional SUSY-pseudo-Hermiticity without separation of variables

We study SUSY-intertwining for non-Hermitian Hamiltonians with special emphasis to the two-dimensional generalized Morse potential, which does not allow for separation of variables. The complexified methods of SUSY-separation of variables and two-dimensional shape invariance are used to construct particular solutions—both for complex conjugated energy pairs and for non-paired complex energies.     

Lie-algebra contractions and separation of variables. Three-dimensional sphere

The Inönü-Wigner contraction from the SO(4) group to the Euclidean E(3) group is used to relate the separation of variables in Helmholtz equations for two corresponding homogeneous spaces. We show how the six systems of coordinates on the three-dimensional sphere contracted to nine systems of coordinates on Euclidean space. As a consequence of the Inönü-Wigner contraction we also consider contractions of the integrals of motion.     

Harmonics on hyperspheres,separation of variables and the bethe ansatz

A class of integrable evolution systems in the spacetimeR ²ⁿ⁺¹ (n 2) based on the generalized self-dual Yang-Mills equations are constructed. It is proved that the Darboux matrix method is applicable to these systems and a lot of explicit solutions are obtained. Starting with the trivial solutions, single soliton solutions and multi-soliton solutions are constructed. They are almost localized and the interaction between solitons is almost elastic.Supported by Chinese national basic research project Nonlinear Science and Japanese Inoue Scientific Foundation.     

Symmetry and separation of variables in second-order differential equations. II

A definition of the complete separation of variables is given, and normal parabolic equations permitting separation of variables in any coordinate system are described; this involves the commutative algebra of differential (not higher than second order) symmetry operators of the equations. In particular, the necessary and sufficient conditions for the complete n separation of variables is formulated in covariant form with respect to arbitrary transformations of the independent variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 7–10, June, 1978.     

Surface operators and magnetic degrees of freedom in yang-mills theories

Magnetic degrees of freedom are manifested through violations of the Bianchi identities and associated with singular fields. Moreover, these singularities should not induce color nonconservation. We argue that the resolution of the constraint is that the singular fields, or defects are Abelian in nature. Recently proposed surface operators seem to represent a general solution to this constraint and can serve as a prototype of magnetic degrees of freedom. Some basic lattice observations, such as the Abelian dominance of the confining fields, are explained then as consequences of the original non-Abelian invariance. Generically, the properties of the two-dimensional defects associated with the surface operators are close to the predictions of the dual models for the magnetic D2 branes.     

Method of complete separation of complex variables in the Hamilton-Jacobi equation

A generalization of the Stekkel spaces is presented which allows a complete separation of complex variables in the Hamilton-Jacobi equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 23–27, September, 1988.     

为什么量子力学中力学量要用厄米算符表示

立足于量子力学的数学表述,详细讨论了厄米算符的4个性质:线性,本征态的完备性,本征态的正交性和本征值的实数性.对于每个性质都给出了正面和反面的讨论,从而揭示了它们各自所代表的意义.     

Separation of variables and symmetry operators for the conformally invariant Klein-Gordon equation on curved spacetime

The separability of the conformally invariant Klein-Gordon equation and the Laplace-Beltrami equation are contrasted on two classes of Petrov type D curved spacetimes, showing that neither implies the other. The second-order symmetry operators corresponding to the separation of variables of the conformally invariant Klein-Gordon equation are constructed in both classes and the most general second-order symmetry operator for the conformally invariant Klein-Gordon operator on a general curved background is characterized tensorially in terms of a valence two-symmetric tensor satisfying the conformal Killing tensor equation and further constraints.     

An invariant formula for a star product with separation of variables

We give an invariant formula for a star product with separation of variables on a pseudo-Kähler manifold.     

A separation property for magnetic Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression L=_ΔA+q$L={\Delta }_A+q$ on a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$, where _ΔA${\Delta }_A$ is the magnetic Laplacian on M$M$ and q≥0$q\ge 0$ is a locally square integrable function on M$M$. In the terminology of W.N. Everitt and M. Giertz, the differential expression L$L$ is said to be separated in L²(M)$L^2\left(M\right)$ if for all u∈L²(M)$u\in L^2\left(M\right)$ such that Lu∈L²(M)$Lu\in L^2\left(M\right)$, we have qu∈L²(M)$qu\in L^2\left(M\right)$. We give sufficient conditions for L$L$ to be separated in L²(M)$L^2\left(M\right)$.