Geodesic
The spectral function asymptotics for second order elliptic differential operator
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 10, Tome 89 (1979), pp. 152-203

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She article is concerned with the problem of the Riesz means of the spectral function for second order elliptic boundary value problem, the boundary being supposed geodesically concave. Asymptotic formulas for the Riesz means with uniform remainder estimate is obtained in the terms of appropriate Green's function asymptotics. An explicit representation of the spectral function in a neighbourhood of the diagonal is valid, this formula yielding a sharp asymptotic remainder estimate for eigenvalues of the regarding boundary value problem. 
@article{ZNSL_1979_89_a12,
     author = {Ya. V. Kurylev},
     title = {The spectral function asymptotics for second order elliptic differential operator},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {152--203},
     publisher = {mathdoc},
     volume = {89},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_89_a12/}
}
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Ya. V. Kurylev. The spectral function asymptotics for second order elliptic differential operator. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 10, Tome 89 (1979), pp. 152-203. http://geodesic.mathdoc.fr/item/ZNSL_1979_89_a12/