Quoting Eugene Salamin <gene_salamin@yahoo.com>:
I had come to the same conclusion myself. If a simple cosine potential leads to complicated Mathieu function solutions, then the appearance of Mathieu functions in tr(M) seems unavoidable, and the full machinery of solving the differential equation should be required just to bring these Mathieu functions into existence.
Are Mathieu functions such bad little critters? Mainly they just don't get used much unless you specialize in the things we are talking about. They´re just lumpy sines and cosines.
However, depending on your potential, you may be able to develop a good approximation and [...] I've done this for other equations, but never tried it in the Mathieu environment.
Could you elaborate a bit more about this "splitting"?
It is described in some notes on complex variable theory which I am using in my class. Look at http://delta.cs.cinvestav.mx/mcintosh/oldweb/pothers.html and then at the first item, ¨Complex Analysis.¨ Section 8 goes into systems of linear differential equations, while section 8.4.3 talks about the sum rule for logarithms. This is somewhat related to the Campbell-Hausdorff formula; W. Magnus has a very nice commutator identity which isn't discussed there. Further down, "Resonance in the Dirac Harmonic Oscillator" splits off the mass from the rest of the Hamiltonian, and was the most interesting of the "other equations" which I mentioned. Still further along, "Periodic Potentials in One Dimension" talke about some Mathieu-like situations, including the use of the Dirac equation. All of these items have options to either just look at them, or to coly them.
Assuming I can't avoid solving the differential equation, rewrite it as a first order matrix differential equation. Let U(x) be the column vector [u(x) u'(x)]. Then U'(x) = K(x) U(x) with
K(x) = [0 0] [ <--- 1,1 element is 1] [V(x)-E 1].
The wave function can be written symbolically as U(x) = G(x) U(0), but because the commutator [K(x1),K(x2)] is nonzero, the expression of G(x) in terms of V(t), 0<=t<=x, is nontrivial. Indeed, we know that if V(x) is a cosine, G(x) has Mathieu functions.
This show up in my treatment where G is written as a sum, the nice half is solved, but then it must be used to transform the second half. Sometimes the process can be repeated, usually not. That is, it always can be, but the result may not be pleasant. With the Dirac Harmonic Oscillator you get some nice spirals.
If V(x) is approximated by a step function, then we can integrate over each step. U(x2) = G0(x2-x1) U(x1), with [...]
Authors have suffested this, I have done it myself. Even though the half intervals are readily soluble, you may need more intervals, and then the hyperbolic trigonometry becomes oppressive, and the convergence may not be as good as for other methods. But I don't know than anyone has made a really systematic study. It works well for the Kronig-Penny combs, where your lattice is made up of delta functions, and that is good enough to deduce bands. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos