The asymptotic behavior of the spectral function for elliptic operators in an unbounded region

We consider elliptic self-adjoint differential operators L of order 2m in a bounded region D? R_n. An asymptotic formula for the function N(λ) = \(N(\lambda ) = \sum\limits_{\lambda _n< \lambda } 1 \) the number of eigenvalues of the operator L less than A. is proved: $$N(\lambda ) = M_0 \lambda ^{n/2m} + o(\lambda ^{n/2m} )$$ whereλ → + ∞ and M₀ is the following constant: $$M_0 = \frac{{V_D }}{{(2\pi )^n \Gamma (1 + n/2m)}}\int_{R_n } {e^{ - L(s)} ds} .$$      

Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters

We consider the problem for eigenvalues of a perturbed two-dimensional oscillator in the case of a resonance frequency. The exciting potential is given by a Hartree-type integral operator with a smooth self-action potential. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundary of spectral clusters, which form around energy levels of the nonperturbed operator. To calculate them, we use asymptotic formulas for quantum means.     

The asymptotic behavior of the discrete spectrum generated by the radial confluent Heun equation with close singularities

Against the background of one of the authors’ (S. Yu. Slavyanov’s) reminiscences of A. A. Bolibrukh, the asymptotic behavior of the spectral curves generated by the boundary-value problem for the confluent Heun equation (that is, an equation with two regular and one irregular singularity) is considered. The spectral curves are constructed for small values of one parameter (the distance between the regular singular points) depending on another parameter, which has the meaning of total charge in physics. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters

We consider an eigenvalue problem for the two-dimensional Hartree operator with a small parameter at the nonlinearity. We obtain the asymptotic eigenvalues and the asymptotic eigenfunctions near the upper boundaries of the spectral clusters formed near the energy levels of the unperturbed operator and construct an asymptotic expansion around the circle where the solution is localized.     

The spectral bounds of the discrete Schrödinger operator

Let H(λ)=−Δ+λb be a discrete Schrödinger operator on ?²(Zd) with a potential b and a non-negative coupling constant λ. When b≡0, it is well known that σ(−Δ)=[0,4d]. When b?0, let and be the bounds of the spectrum of the Schrödinger operator. One of the aims of this paper is to study the influence of the potential b on the bounds 0 and 4d of the spectrum of −Δ. More precisely, we give a necessary and sufficient condition on the potential b such that s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar necessary and sufficient condition on the potential b such that M(−Δ+λb) is lower than 4d for λ small enough. In dimensions d=1 and d=2, the situation is more precise. The following result was proved by Killip and Simon (2003) (for d=1) in [5], then by Damanik et al. (2003) (for d=1 and d=2) in [3]:

     

On the asymptotic behavior of the discrete spectrum of the Dirichlet and Neumann problems for the Laplace-Beltrami operator on a regular polyhedron in the Lobachevskii space

M. V. Lomonosov Moscow State University. Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 24, No. 1, pp. 21–28, January–March, 1990.     

The information operator and the asymptotic behavior of the toeplitz determinant

An asymptotic representation of the information operator, corresponding to a stationary Gaussian vector sequence, is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 179–185, 1988.     

Semiclassical asymptotic spectrum of the two-dimensional Hartree operator near a local maximum of the eigenvalues in a spectral cluster

Theoretical and Mathematical Physics - We consider the eigenvalue problem for the two-dimensional Hartree operator with a small nonlinearity coefficient. We find the asymptotic eigenvalues and...     

The discrete spectrum in the gaps of a second-order perturbed periodic operator

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 89–92, April–June, 1991.     

Remainder estimates in spectral asymptotic formulas for linear operator pencils

One gives a qualified remainder estimate in the asymptotic formulas of the distribution functions of the positive and negative spectra of a linear pencil of differential operators, where the lowest operator is not assumed to be elliptic.Translated from Problemy Maternaticheskogo Analiza, No. 9, pp. 34–56, 1984.     

Asymptotic behavior of the discrete spectrum of the Schrödinger operator in electric and homogeneous magnetic fields

The asymptotic behavior of bound states is considered to the left of the boundary of the essential spectrum of the Schrödinger operator in homogeneous magnetic and decreasing electric fields. The electric potential is not considered nonpositive. It is assumed that the integral of the potential along the direction of the magnetic field has a powerlike behavior at infinity. It is shown that the asymptotic behavior of the bound states is powerlike and its leading term is computed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol.182, pp. 131–141, 1990.