The method of approximate spectral projection.

*(English. Russian original)*Zbl 0614.35021
Math. USSR, Izv. 27, 451-502 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 6, 1177-1228 (1985).

The method of approximate spectral projection [see V. N. Tulovskij and M. A. Šubin, Mat. Sb., Nov. Ser. 92(134), 571-588 (1973; Zbl 0286.35059)] is further developed. The improvement includes a better estimation of the remainder in the problem of the type \(Au=tBu\) (A is a pseudodifferential operator of elliptic type in the sense of Douglis- Nirenberg, B is a subordered operator and one of the operators A, B is positive definite). The pseudodifferential operators are calculated similarly as by Weyl-Hörmander [see L. Hörmander, Commun. Pure Appl. Math. 32, 359-443 (1979; Zbl 0388.47032)] only in this article the operator order is determined by two operator-functions and not by one scalar function. Essential theorems are proved in the first part. In the second part these theorems are used under conditions of the Dirichlet type to study following problems: spectral problems of the shell theory, problems of the \(Au=tBu\) type, spectral asymptotics of pseudodifferential operators, asymptotics of the discrete spectrum becoming dense at the lower boundary of real, scalar and matrix Schrödinger operators perturbed by differential operators of the first order.

Reviewer: V.Burjan

##### MSC:

35J10 | Schrödinger operator, Schrödinger equation |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

35S15 | Boundary value problems for PDEs with pseudodifferential operators |

47A10 | Spectrum, resolvent |

74K25 | Shells |

74H45 | Vibrations in dynamical problems in solid mechanics |