Geodesic
Two-sided estimates for eigenvalues of the Schr\"odinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 24 (1975) no. 3, pp. 412-418

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Schrödinger equation is substituted into two systems of $n$ linear equations with $n$ unknown quantitys. The coefficients of these equations are the matrix elements of the hamiltonian between quasiclassical wavefunctions. The solution of these systems give the two-side estimates for eigenvalues of the Schrodinger equation. The relative distance between boundary $\simeq\lambda^k$, where $\lambda$ is the parameter of the quasiclassical decomposition for the $n$-th wavefunction, $k$ is the number of terms in this decomposition. 
@article{TMF_1975_24_3_a12,
     author = {G. V. Ryazanov},
     title = {Two-sided estimates for eigenvalues of the {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {412--418},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1975_24_3_a12/}
}
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G. V. Ryazanov. Two-sided estimates for eigenvalues of the Schr\"odinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 24 (1975) no. 3, pp. 412-418. http://geodesic.mathdoc.fr/item/TMF_1975_24_3_a12/