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Asymptotic solutions for Hartree equations and logarithmic obstructions for higher corrections of semiclassical approximation. (English) Zbl 1123.81348

Danilin, A. R. (ed.), Asymptotic expansions, approximation theory, topology. Transl. from the Russian. Moscow: Maik Nauka / Interperiodica. Proceedings of the Steklov Institute of Mathematics 2003, Suppl. 1, S123-S128 (2003).
From the introduction: We consider the eigenvalue problem for a nonlinear Hartree equation in a 3-dimensional space: \[ (-\varepsilon^2\Delta+|x|^2) \psi(x)-\psi(x) \int_{\mathbb R^3} \frac{|\psi(x')|^2}{|x-x"|}\, dx'= \Lambda\psi(x),\tag{1} \]
\[ \|\psi\|_{L^2}=1, \tag{2} \] where \(\Delta\) is the Laplace operator, and \(\varepsilon\) is a small parameter. A Hartree equation with the singular Coulomb interaction kernel is one of the most common models used in physics of a selfconsistent field of charged particles (as well as nucleons). When studying the asymptotics of the spectrum of this equation the primary role is played by the singularity of the interaction kernel. This prevents applying any well-known “smooth” semiclassical approximation methods and implies absolutely different behavior of asymptotics for the spectrum due to adding fractional powers of \(\varepsilon\) and logarithmic corrections into the quantization rule.
This paper discusses steps for constructing a sequence of asymptotic eigenvalues \(\Lambda\) in the problem (1), (2) and the corresponding eigenfunctions \(\psi\) whose supports up to \(O(\varepsilon^\infty)\) are 2-dimensional discs (annuli) \(D=\{(z,\rho,\varphi)\mid z=0\), \(\rho_-\leq\rho\leq\rho_+\}\) (\((z,\rho,\varphi)\) denots cylindrical coordinates for the points \(x\in\mathbb R^3\)).
In addition we discuss difficulties arising in problems (1), (2) while constructing eigenfunctions localizsed near 1-dimensional segments \[ L=\{(z,\rho,\varphi)\mid \rho=0,\;z_-\leq z\leq z_+\}.\tag{3} \]
For the entire collection see [Zbl 1116.35001].

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
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