Admits Separation Of Variables Research Articles

Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

Algebra of Symmetry Operators and Integration of the Klein–Gordon Equation in an External Electromagnetic Field

The structure of the algebra of first-order differential symmetry operators of the Klein-Gordon equation in an external electromagnetic field is investigated from positions of the theory of cohomologies of Lie algebras. A method for integrating the given equation is proposed, based on noncommutative reduction, and, as a consequence, the corresponding integrability condition is obtained. An example of an integrable Klein- Gordon equation in a spacetime with a Steckel metric is considered, which does not admit separation of variables when an electromagnetic field is switched on. tensor of which is invariant with respect to the group of motions of the spacetime metric. The goal of this work is to describe the structure of the algebra of first-order symmetry operators of the equation under consideration from positions of the theory of cohomologies of Lie algebras, and also to propose a method of integration using the indicated symmetry algebra. An integrability condition for the Klein-Gordon equation in an electromagnetic field is derived, expressed in terms of the dimensionality of the Lie algebra of the group of motions of the metric and the so-called cohomology index of this algebra, this index depending on the class of cohomologies defined by the electromagnetic field. Note that in contrast to the well-known method of separation of variables, the integration method proposed in this work is based on the idea of noncommutative reduction (1), which makes more effective use of the algebra of first-order differential symmetry operators. In a number of cases, this eliminates the necessity of investigating hidden symmetries of the problem, associated with Killing tensors of second rank (and higher), which must often be done in the case of separation of variables (2, 3). To wrap things up, this paper considers an example of the Klein-Gordon equation in spacetime with a four- dimensional group of motions of the metric, which is integrable with the help of the approach proposed here. An interesting qualitative aspect of the example is the possibility of separation of variables in this equation in the absence of an external electromagnetic field and the impossibility of doing so in its presence.

The phase retrieval problem: a spectral parameter power series approach

This paper presents the application of the spectral parameter power series method [Pauli, Math Method Appl Sci 33:459–468 (2010)] for constructing the Green’s function for the elliptic operator $$-\nabla \cdot I\nabla $$ in a rectangular domain $$\varOmega \subset \mathbb R ^{2}$$ , where $$I$$ admits separation of variables. This operator appears in the transport-of-intensity equation (TLE) for undulatory phenomena, which relates the phase of a coherent wave with the axial derivative of its intensity in the Fresnel regime. We present a method for solving the TIE with Dirichlet boundary conditions. In particular, we discuss the case of an inhomogeneous boundary condition, a problem that has not been addressed specifically in other works, under the restricted assumption that the intensity $$I$$ admits separation of variables. Several simulations show the validity of the method.

Exactly solvable two-dimensional complex model with a real spectrum

Using supersymmetric intertwining relations of the second order in derivatives, we construct a two-dimensional quantum model with a complex potential for which all energy levels and the corresponding wave functions are obtained analytically. This model does not admit separation of variables and can be considered a complexified version of the generalized two-dimensional Morse model with an additional sinh −2 term. We prove that the energy spectrum of the model is purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. We explicitly find the symmetry operator, describe the biorthogonal basis, and demonstrate the pseudo-Hermiticity of the Hamiltonian of the model. The obtained wave functions are simultaneously eigenfunctions of the symmetry operator.

Classification of F algebras and noncommutative integration of the Klein-Gordon equation in Riemannian spaces

The method of noncommutative integration of linear differential equations [A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 4, 116; No. 5, 100 (1991)] is used to integrate the Klein-Gordon equation in Riemannian spaces. The situation is investigated where the set of noncommuting symmetry operators of the Klein-Gordon equation consists of first-order operators and one second-order operator and forms a so-called F algebra, which generalizes the concept of a Lie algebra. The F algebra is a quadratic algebra in the given situation. A classification of four- and five-dimensional F algebras is given. The integration of the Klein-Gordon equation in a Riemannian space, which does not admit separation of variables, is demonstrated in a nontrivial example.

Symmetry and separation of variables in a linear differential equation of second order. I

The definition of partial Π- separation of variables is given and a covariant description is carried out of nonparabolic equations which in some coordinate system admit separation of variables; to this end, commutative algebras of differential symmetry operators of the equation of no higher than second order are used. In particular, necessary and sufficient conditions for complete Π-separation of variables are formulated in covariant form with respect to arbitrary transformations of the independent variables.