I have a matrix $M(\text{kx$\_$},\text{ky$\_$})=\left( \begin{array}{cc} a_2 (1-\cos (\text{kx}) \cos (\text{ky}))+a_1 (1-\cos (\text{kx})) & 2 a_2 \sin (\text{kx}) \sin (\text{ky}) \\ 2 a_2 \sin (\text{kx}) \sin (\text{ky}) & a_2 (1-\cos (\text{kx}) \cos (\text{ky}))+a_1 (1-\cos (\text{ky})) \\ \end{array} \right)$
M[kx_, ky_] = ({
{Subscript[a, 1] (1 - Cos[kx]) +
Subscript[a, 2] (1 - Cos[kx] Cos[ky]),
2 Subscript[a, 2] Sin[kx] Sin[ky]},
{2 Subscript[a, 2] Sin[kx] Sin[ky],
Subscript[a, 1] (1 - Cos[ky]) +
Subscript[a, 2] (1 - Cos[kx] Cos[ky])}
});
I want to find the eigenvalues and eigenvectors at the limit of kx$\to$0 and ky$\to$0 in any direction of $\vec k$.
If I define ky=0
In[57]:= Eigensystem[M[kx, 0]]
Out[57]= {{(-1 + Cos[kx]) (-Subscript[a, 1] - Subscript[a,
2]), -(-1 + Cos[kx]) Subscript[a, 2]}, {{1, 0}, {0, 1}}}
Eigenvalue is $(a_1+a_2) (1-\cos (\text{kx})),a_2 (1-(\cos (\text{kx}))$
In the limit, it is $(a_1+a_2)\text{kx}^2/2,a_2 \text{kx}^2/2$
If I define kx=0
In[58]:= Eigensystem[M[0, ky]]
Out[58]= {{(-1 + Cos[ky]) (-Subscript[a, 1] - Subscript[a,
2]), -(-1 + Cos[ky]) Subscript[a, 2]}, {{0, 1}, {1, 0}}}
Eigenvalue is $(a_1+a_2) (1-\cos (\text{ky})),a_2 (1-(\cos (\text{kx}))$
In the limit, it is $(a_1+a_2) \text{ky}^2/2,a_2\text{kx}^2/2$
The problem occurs when I series expand Eigenvalue of $M(\text{kx},\text{ky})$
In[61]:= Series[Eigensystem[M[kx, ky]], {kx, 0, 2}, {ky, 0, 2},
Assumptions -> {Subscript[a, 1] > 0, Subscript[a, 2] > 0, kx > 0,
ky > 0}]
It gives eigenvalues
$\left(\frac{a_2 \text{ky}^2}{2}+O\left(\text{ky}^3\right)\right)+\text{kx}^2 \left(\left(-\frac{8 a_2^2}{a_1}+\frac{a_2}{2}+\frac{a_1}{2}\right)+\left(\frac{2 a_2^2}{a_1}-\frac{a_2}{4}\right) \text{ky}^2+O\left(\text{ky}^3\right)\right)+O\left(\text{kx}^3\right)$
and
$\left(\left(\frac{a_1}{2}+\frac{a_2}{2}\right) \text{ky}^2+O\left(\text{ky}^3\right)\right)+\text{kx}^2 \left(\left(\frac{8 a_2^2}{a_1}+\frac{a_2}{2}\right)+\left(-\frac{2 a_2^2}{a_1}-\frac{a_2}{4}\right) \text{ky}^2+O\left(\text{ky}^3\right)\right)+O\left(\text{kx}^3\right)$
It agree with previous answer in the direction of kx=0 but disagree in the direction of ky=0
If I do series expansion with the order of $\{\text{ky},0,2\},\{\text{kx},0,2\}$, it becomes opposite.
What causes this bug and how can I get the correct answer?
Update I rewrite to matrix in polar coordinate k and $\theta_k$ and series expand only to k. It is correct this time, but problem still remain in cartesian coordinate.