wrote:

Dear all,

Happy new year!

I'd like to get a clear concept of Debye potentials.
For the sake of this, I searched around the internet and
checked several classic textbooks, like Jackson's and
Stratton's, but no satisfactory results. Instead I get
several papers describing Debye potentials published
decades before ("Debye potential representation of
vector fields").

From those papers I find out that:
Debye potentials have something to do with the special
case of Helmholtz Theorem with divergenceless vector
fields. It's proved then this field can be represented by
two scalar potentials:
F = Lø + curl(L=÷),
where F is the vector field and L is the standard orbital
angular momentum operator. It's said these two scalar
potentials are Debye potentials. (Is this obsolete? Why
isn't there any like content in today's textbooks)

The name of Debye is usually not associated with this decomposition.
Instead you can find the fields derived from the potentials
described as toroidal and poloidal fields.
They play a role in the theory of for example
the earth's dynamo, the sun's magnetic field,
or magnetically confined fusion devices.

Except this I also get various descriptions, but I can't
figure out a unified idea. Could anyone suggest some
detailed reading?

BTW, it seems that Debye potentials have close
relation with multipole expansion. Is this true and what's
that?

The basic toroidal fields are sigular on the Z-axis,
and the multipole expansion gets replaced by the same thing
with the Legendre polynomials Plm
replaced by the Legendre functions of the second kind,
Qlm.
Since the Legendre functions of the second kind
lack the nice orthoganality properties
of the corresponding Legendre polynomials
this multipole expansion of the second kind is less useful.

Best,

Jan