Orthogonal and non-orthogonal separation of variables in the wave equation u tt -u xx +V(x)u=0u tt -u xx +V(x)u=0

Journal of Mathematical Physics

1999

We classify (1+3)-dimensional Schrödinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get eleven classes of the electromagnetic vector potentials of the electromagnetic field A(t, x) = (A 0 (t, x), A(t, x)) providing separability of the corresponding Schrödinger equations. It is established, in particular, that the necessary condition for the Schrödinger equation to be separable is that the magnetic field must be independent of the spatial variables. Next, we prove that any Schrödinger equation admitting variable separation into second-order ordinary differential equations can be reduced to one of the eleven separable Schrödinger equations mentioned above and carry out variable separation in the latter. Furthermore, we apply the results obtained for separating variables in the Hamilton-Jacobi equation.

Section: Lemmamentioning

confidence: 99%

On separable Schrödinger equations

Zhdanov

Journal of Mathematical Physics

1999

“…The principal aim of the present paper is to apply the direct approach to variable separation in PDEs suggested in [7]- [9] to solve KE. As is wellknown, separability of PDE is intimately connected to its symmetry within the class of second-order differential operators [10].…”

Section: Introductionmentioning

confidence: 99%

“…Exact solutionsRemarkably, for the equation under study it is possible to give a complete account of solutions with separated variables. For the case when KE separates into three first-order ODEs(7), we get the following family of its exact…”

mentioning

confidence: 99%

Separation of variables in the Kramers equation

Zhdanov

1999

J. Phys. A: Math. Gen.

“…Our analysis is based on the direct approach to variable separation in linear PDEs suggested in [7]- [9]. It has been successfully applied to solving variable separation problem in the wave [7] and Schrödinger equations [8]- [12] with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…It has been successfully applied to solving variable separation problem in the wave [7] and Schrödinger equations [8]- [12] with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

On separable Fokker-Planck equations with a constant diagonal diffusion matrix

1999

J. Phys. A: Math. Gen.

We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients B 1 ( x), B 2 ( x), B 3 ( x) providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients B( x) must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructed.