How can I fit a matrix function with multiple variables to given eigenvalues?

Question

I have a 5x5 matrix function H[kx]

H[kx_]:={{T1, kx T8, -I T6, kx T9, -I kx T10}, 
 {kx T8, T2, -I kx T11, kx T12, I kx T13}, 
 {I T6, I kx T11, T3, -I kx T14, kx T15}, 
 {kx T9, kx T12, I kx T14, T4, -I T7}, 
 {I kx T10, -I kx T13, kx T15, I T7, T5}
}

with real parameters T1 to T15, and I have data as follow

dat={{-0.005, -0.418971, 1.24115, 2.6321, 3.07757, 4.40013}, 
 {-0.004, -0.418735, 1.24086, 2.63201, 3.07785, 4.40028}, 
 {-0.003, -0.418551, 1.24063, 2.63194, 3.07806, 4.4004}, 
 {-0.002, -0.41842, 1.24047, 2.63189, 3.07822, 4.40048}, 
 {-0.001, -0.418342, 1.24038, 2.63185, 3.07831, 4.40053}, 
 {0., -0.418316, 1.24034, 2.63184, 3.07834, 4.40055}, 
 {0.001, -0.418342, 1.24038, 2.63185, 3.07831, 4.40053}, 
 {0.002, -0.41842, 1.24047, 2.63189, 3.07822, 4.40048}, 
 {0.003, -0.418548, 1.24063, 2.63194, 3.07807, 4.4004}, 
 {0.004, -0.418728, 1.24085, 2.63201, 3.07785, 4.40028}, 
 {0.005, -0.418959, 1.24114, 2.63211, 3.07758, 4.40012}
}

For each i, kx=dat[[i,1]], and dat[[i,2;;6]] are the eigenvalues (from small to big) which will be fitted by H[kx].

The aim is to find all the parameters T1 to T15 which best produce the eigenvalues given by dat for all kx. How can I achieve this?

Answer 1

Not having good starting values at hand, nor sufficient time to spare with your problem, here was the best I could do:

{mv, am} = 
 Quiet @ NMinimize[Norm[Norm[
   Map[Function[x, With[{kx = First[#]}, CharacteristicPolynomial[H[kx], x]] // Evaluate],
       Rest[#]]] & /@ dat],
                {T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12, T13, T14, T15},
                Method -> {"NelderMead", RandomSeed -> 4}]
   {0.020097 + 0. I,
    {T1 -> 1.61337, T2 -> -0.418315, T3 -> 2.70502, T4 -> 3.47363,
     T5 -> 3.55875, T6 -> 0.73906, T7 -> 0.883149, T8 -> -5.58534,
     T9 -> 2.15443, T10 -> -1.25767, T11 -> 3.13062, T12 -> -1.82002,
     T13 -> 1.77503, T14 -> -0.694595, T15 -> -0.768617}}

Even this feeble attempt yields something usable:

mf[kx_] = H[kx] /. am;
Eigenvalues[mf[First[#]]] - Sort[Rest[#], GreaterEqual] & /@ dat

gives

{{0.00027062, 0.000618862, -0.00025875, -0.000162659, 4.80406*10^-6},
 {0.000107527, 0.000256957, -0.000105524, -0.0000752058, 3.12297*10^-6},
 {-0.0000226674, -0.0000167649, 0.0000136659, -2.74297*10^-6, 1.38639*10^-6},
 {-0.000109957, -0.00022229, 0.0000988083, 0.0000447379, 5.78009*10^-7},
 {-0.000164336, -0.000339608, 0.000159896, 0.0000672356, 6.90641*10^-7},
 {-0.000185798, -0.000378715, 0.000176925, 0.0000847389, 7.26231*10^-7},
 {-0.000164334, -0.000339608, 0.000159897, 0.000067227, 6.95853*10^-7},
 {-0.000109939, -0.00022229, 0.0000988172, 0.0000446694, 6.19705*10^-7},
 {-0.0000226064, -0.0000267654, 0.0000136958, -2.97409*10^-6, -1.47291*10^-6},
 {0.000107672, 0.000256956, -0.000105453, -0.0000657536, -3.5436*10^-6},
 {0.000280902, 0.00060886, -0.000268611, -0.000153729, -6.54493*10^-6}}

Someone with more time to play around with the Method option of NMinimize[] should be able to find better results than mine...