1. The review is mainly devoted to the presentation of results on asymptotics in the spectral parameter of a discrete spectrum of self-adjoint differential operators, mainly partial differential operators. The papers reviewed in the Russian Academy of Sciences "Mathematics" in 1967-1975 are discussed. Earlier articles are mentioned in cases where it is necessary in the course of the presentation. The status of the issue for 1967 can be found in Clark's review. We can also recommend reviews by R. A. Alexandryan, Y. M. Berezansky, V. A. Ilyin and A. G. Kostyuchenko and Y. M. Berezansky, V. I. Gorbachuk and M. L. Gorbachuk (RZMat, 1976, 6B747).
The review does not address specifically non-self-adjoint issues, as well as quasi-related numbers (spectrum on a "non-physical sheet").
Our presentation is concentrated around the following issues:
1) The form of the main asymptotic term for N(X) (spectrum distribution function) in various problems (§ 1). Figuring out the limits to which the "standard" expressions for the main term are valid (§§ 2, 3, 4). Finding the type of asymptotics in cases where these expressions are unsuitable (§§ 6, 7).
2) Estimation of the remainder in the asymptotic formula (§ 8).
3) Investigation of the following terms of spectral asymptotics for N(1) (§ 9). Any complete results are obtained here (and are well known) only in one-dimensional problems. In the multidimensional case, appropriately averaged (smoothed) asymptotic expansions are discussed.
4) Problems on the asymptotics of N(K) with respect to a small parameter included in the equation (§ 5).
Some issues that are loosely related to the main text of the review are included in section 10.
We almost do not touch on some issues of self-adjoint theory related to the asymptotics of the spectrum. Among them: decompositions by eigenfunctions*); trace formulas; problems with nonlinear parameter occurrence (bundles); problems with a small parameter in the spirit of the famous works of M. I. Vishik and L. A. Lusternik; the spectrum of the Schrodinger equation with a random potential**).
The asymptotics of the spectral function are discussed only in cases where it serves as a source of asymptotic formulas for eigenvalues. For more information about the asymptotics of the spectral function, see Hermander.
On the application of asymptotic methods in one-dimensional problems. The multidimensional version of the method developed by V. P. Maslov in KB is described in his book and in the work of V. P. Maslov and M. V. Fedoryuk.
Finally, we will point out books and lecture courses largely devoted to the presentation of spectral asymptotics: M. A. Naimark, Agmon, A. G. Kostyuchenko, M. S. Burman and M. 3. Solomyak.
Methods for studying the asymptotics of the spectrum. Actually, as well as R. Courant et al. We will use the term "variational method" in an extended sense, meaning not only the application of the minimaximal principle, but also the active use of the unitary invariance of the spectrum and the means of perturbation theory of the abstract theory of operators. The variational method is directly applicable only in semi-limited problems. A casa de apostas 1xBet atrai novos jogadores com bônus de 100% no primeiro depósito para registro até R$1200. Receba um bônus de boas-vindas de até R$9500 + 150 rodadas grátis na 1xBet. Para receber o bônus de registro, você deve usar o Código promocional 1xBet Portugal este é o código que ativa todas as promoções e bônus disponíveis em sua conta hoje.Esta forma de incentivo está disponível apenas para jogadores registrados que confirmaram seu número de telefone. A casa de apostas dá as boas-vindas aos novos clientes. A empresa não oferece um bônus sem depósito no momento do registro, mas você pode de depósito.