Geodesic
Regularized traces of boundary problems in case of multiple roots of characteristic polynomial
Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 567-583

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The boundary problem on a segment for differential equation of $n$ order with coefficients polynomially depending on spectral parameter $\lambda$ is considered. In the general case of multiple roots of Tamarkin's characteristic polynomial the regularized traces, i.e. the sums $\sum\limits_k[\lambda_k^m-A_m(k)]$, $m\in\mathbb{N}$, are calculated, where $\lambda_k$ are eigenvalues of the problem, and $A_m(k)$ are totally defined numbers, ensuring the convergence of series. 
@article{FPM_1998_4_2_a7,
     author = {A. S. Pechentsov},
     title = {Regularized traces of boundary problems in case of multiple roots of characteristic polynomial},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {567--583},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a7/}
}
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A. S. Pechentsov. Regularized traces of boundary problems in case of multiple roots of characteristic polynomial. Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 567-583. http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a7/