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Zbl 0919.35095
Petkov, Vesselin; Popov, Georgi
Semi-classical trace formula and clustering of eigenvalues for Schrödinger operators. (English)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 68, No.1, 17-83 (1998). ISSN 0246-0211

Authors' summary: This paper is devoted to certain semi-classical asymptotics of a Schrödinger type operator $A(h)$ in the vicinity of a regular value $E$ of its principal symbol $a_0(x,\xi)$. We investigate the semi-classical behaviour of the number $N_{E+rh,c}(h)$ of all eigenvalues $\lambda_{j}(h)$ of $A(h)$ situated in the interval $[E+rh-ch,E+rh+ch]$, where the energy shift parameter $r$ and the size constant $c>0$ are both bounded. The behaviour of $N_{E+rh,c}(h)$ for small $h$ depends on an oscillating term $Q(h,r)$ which is related to the periodic trajectories of the Hamiltonian vector field $H_{a_0}$ on the energy hypersurface $\Sigma=\{(x,\xi):a_0(x,\xi)=E\}$. If $Q(h,r)$ is uniformly continuous in $r$ for any $0<h\leq h_0$, we obtain asymptotics of the counting function $N_{E+rh,c}(h)$ as $h$ tends to zero. On the other hand, the points of discontinuity of $Q(h,r)$ in $r$ may give rise to a clustering of eigenvalues of $A(h)$ near the energy level $E$. Such jumps of the function $Q$ in $r$ are described in terms of a suitable quantization condition. In particular, if $a_0$ is analytic in a neighborhood of $\Sigma$ and the energy surface is connected and of contact type we obtain a complete description of the asymptotics of $N_{E+rh,c}(h)$. Moreover, we obtain a new semi-classical trace formula giving for any $\rho$ with Fourier transform $\hat{\rho}\in C_0^{\infty}({\Bbb R})$ the asymptotics of $$ \sum_{\lambda_{j}(h)\leq\lambda}\rho\left(\frac{E-\lambda_{j}(h)}{h}\right)$$ in terms of certain dynamical and topological characteristics of the periodic trajectories of $H_{a_0}$ on $\Sigma$ without any additional clean intersection assumptions.
[Marcel Griesemer (Regensburg)]

MSC 2000:: *35P20 Asymptotic distribution of eigenvalues for PD operators
35J10 Schroedinger operator
35P15 Estimation of eigenvalues for PD operators

Keywords: semi-classical asymptotics; periodic trajectories; counting function; quantization

Cited in: Zbl 1113.35331

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