Are There Square Integrable Vector Field Solutions of Curl(F) = F ?
More exactly, consider three functions which satisfy the set of three simultaneous linear first order PDE's implied by the expression curl(F) = F with no finite boundary conditions. These functions for the components of F should be expressed in terms of known functions to provide some intuitive understanding of the space of solutions and provide a basis for proving some theorems about that space. Square integrable (SQI) means the integral of the square of the solution over all space gives a finite value.
Solutions without finite boundary conditions exist in at least three coordinate systems: rectangular, cylindrical and spherical. In rectangular coordinates (x,y,z) the solutions are of the form (0, sin(x), cos(x)). In cylindrical coordinates a solution can be obtained by replacing the sin and cos with first and zero order Bessel functions of the radius respectively. In spherical coordinates, the solutions have been expressed in terms of spherical Bessel functions by Chubykalo and Espinoza in "Journal of Physics A Math 35" in 2002. These solutions goes to zero in all directions but not fast enough to localize the function to get an SQI solution. No solution that I know of is SQI. Are there SQI solutions (with no boundary conditions except at infinity and which can be expressed in terms of known functions) in general orthogonal curvilinear coordinate systems?
Special emphasis is on toroidal and bispherical coordinates. Definitions of these coordinate systems can be found online by using those names as search terms or in "Methods of Theoretical Physics" by Morse and Feshbach and also in "Field Theory Handbook" by Moon and Spencer. As explained by Morse and Feshbach, use of a special coordinare system is a guess that a solution can be found with integral lines that coincide with the coordinate lines thus permitting a simple expression for that solution. I have not been able to find any way to separate the variables in these coordinates for the expression curl(F) = F..
Here is something that may help in finding SQI solutions. Since F is a curl, it must be that div(F) = 0 and its integral lines are closed curves or can terminate only at a singularity or infinity. Thus the location of the singular points of the coordinate system play a strong, possibly dominant, roll in forming the shape of the solution. In the coordinate systems mentioned above there is at most one singular point or at most one singular line.
In other orthogonal curvilinear systems there are more than one singular point and/or more than one singular line which might help to localize the solution. For example, in bispherical coordinates the two foci are singular points and in an infinitesimal neighborhood around them, the metric is the same as spherical coordinates so there are solutions like those of spherical coordinates in the infinitesimal focal neighborhoods. These solutions can be continued (but numeric solutions are of little value in carrying out the objectives mentioned above.) The spheres of fixed value of a larger size which surround a focal point in bispherical coordinates may form a region containing most of the value when they crunch into the corresponding spheres around the other focal point. Thus a more localized solution may be created which might be SQI.
Questions: Is this reasoning correct or are there flaws? Are there solutions F (in terms of known functions) of the expression curl(F) = F for toroidal or bispherical coordinates? Are there SQI solutions F (in terms of known functions and filling all space) of the expression curl(F) = F for toroidal or bispherical coordinates? If so, what are they? Are there reasons for thinking there are no such solutions?