Separation of variables can be a powerful technique for solving many of the partial differential equations that arise in physics contexts. Upper-division physics students encounter this technique in multiple topical areas including electrostatics and quantum mechanics. To better understand the difficulties students encounter when utilizing the separation of variables technique, we examined stud...

The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4,<), whose maximal compact subgroup is the Fourier group U(2) F ; this includes isotropic and anisotropic Fourier transforms, screen rotations and gyrations in the phase space of ray positions and optical momenta. Deforming classical optics into a Hamiltonian system whose positions and momenta...

It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by re-formulating the Ernst equation and its associated linear system in terms of a non-autonomous Schlesinger-type dynamical system. The conformal factor of the metric coincides (up to some explicitly computable factor) with the τ -function of the Ernst equation in the...

The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which gives separation variables for systems for which an R-matrix is known. Previously the FBA was only known for SL(2) and SL(3) (and associated) R-matrices. In this paper I advance Sklyanin's program by giving the FBA for certain systems with SL(N) R-matrices. This is achieved by constructing rational functions A(u) and B(u) ...

We study SUSY −intertwining for non-Hermitian Hamiltonians with complex supercharges. Special emphasis is given to the two-dimensional generalized (singular) Morse potential. This model does not allow for separation of variables nevertheless a part of its spectrum and eigenfunctions can be constructed. The spectrum includes both complex conjugated energy pairs and non-paired complex energies fo...

Abstract. We extend Sklyanin’s method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are a...

A signature independent formalism is created and utilized to determine the general second-order symmetry operators for Dirac’s equation on two-dimensional Lorentzian spin manifolds. The formalism is used to characterize the orthonormal frames and metrics that permit the solution of Dirac’s equation by separation of variables in the case where a second-order symmetry operator underlies the separ...

A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions in terms of Abelian functions of Kleinian type (i.e. the higher-genus analogues of the Weierstrass elliptic functions) is discussed as well as the connections...

we observe that the variable φ enters the expression in the form of the operator ∂2/∂φ2. This operator—1D Laplace operator with respect to the variable φ—is a Hermitian operator in the space of single-valued functions of the angle φ, because single-valuedness is equivalent to the 2π-periodicity, and Laplace operator is Hermitian in the space of periodic functions. Hence, we have ONB of the eige...