Variable separation for Heat and Schrödinger equations

Abstract

All R-separable coordinate systems are classified for the equation Δₘ Ψ + 2ε ∂/∂t Ψ = EΨ (*) where Δₘ = ᵐΣ ᵤ₌₁ ∂²/(∂yᵘ)² and the variables t and yᵘ are real. An R-separable coordinate system for (*) is a coordinate system yᵘ = yᵘ (xᵏ), t = t (xᵏ) such that (*) admits a solution of form Ψ = eᴿ⁽ˣⁱ⁾ ᵐ⁺¹Πᵢ₌₁ Ψᵢ(xⁱ;c₁, ···, cₘ₊₁). For ε = ⁱ/₂ and ε = - ¹/₂, E = 0, (*) yields standardised versions of the Schrӧdinger and Heat equations respectively. Recognition of (*) as a symmetry-reduced version of the Helmholtz equation on m + 2 dimensional Minkowski space, for which there is a well developed theory of separation, is central to the solution of the classification problem. The solution is built from the case ε = 0 which has been solved by Kalnins and Miller in an article submitted to the Transactions of the American Mathematical Society. We find the operators and separation equations for (*) and give a detailed treatment of the physically interesting case m =3. The operators are always in the enveloping algebra of the Schrӧdinger algebra. This means that much of the special function theory relating to the separated solutions can be reduced to problems in the representation theory of this algebra. Various complex extensions of (*) are treated. In particular a class of systems is uncovered leading to nontrivial R-separation for the Hamilton-Jacobi equation corresponding to (*). A formula is derived for expressing the second order Killing tensors describing separation in flat spaces in terms of the enveloping algebra. This formula forms the basis of a program written in the symbolic language MACSYMA capable of producing all the time consuming details of separation.