Local trace asymptotics in the self-generated magnetic field

We consider a semiclassical asymptotics of local trace for the 3D-Schroedinger operator with self-generated magnetic field; it is given by Weyl expression with O(h^{-1}) error and under standard condition to Hamiltonian trajectories even o(h^{-1}). In comparison to v1,2 errors are corrected, new results are added and more details are provided. Misprint correction in comparison to v3Comment: 24 p

Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators

Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function $e_{h,\varepsilon}(x,x,\lambda)$ for a scalar operator \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} where $A^0$ is an elliptic operator and $B(x,hD)$ is a periodic or almost periodic perturbation. In particular, a complete semiclassical asymptotics of the integrated density of states also holds. Further, we consider generalizations.Comment: 24p

Magnetic Schroeodinger Operator: Geometry, Classical and Quantum Dynamics and Spectral Aymptotics

I study the Schroedinger operator with the strong magnetic field, considering links between geometry of magnetic field, classical and quantum dynamics associated with operator and spectral asymptotics.Comment: 23 p

Complete Differentiable Semiclassical Spectral Asymptotics

For an operator $A:= A_h= A^0(hD) + V(x,hD)$ with a "potential" $V$ decaying as $|x|\to \infty$ we establish under certain assumptions the complete and differentiable with respect to $\tau$ asymptotics of $e_h(x,x,\tau)$ where $e_h(x,y,\tau)$ is the Schwartz kernel of the spectral projector.Comment: 12p

Sharp Spectral Asymptotics for four-dimensional Schroedinger operator with a strong magnetic field. II

I consider 4-dimensional Schr\"odinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate $O(\mu^{-1}h^{-3})$ and the principal part $\asymp h^{-4}$ where $h\ll 1$ is Planck constant and $\mu \gg 1$ is the intensity of the magnetic field. For general potentials remainder estimate $O(\mu^{-1}h^{-3}+\mu^2h^{-2})$ is achieved.Comment: 57p

The geometry of the Weil-Petersson metric in complex dynamics

In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion. For the upper bound, we estimate the intersection of a circle $S_r = \{z : |z| = r\}$, $r \approx 1$, with an invariant subset $\mathcal G \subset \mathbb{D}$ called a half-flower garden, defined in this work. For the lower bound, we use gradients of multipliers of repelling periodic orbits on the unit circle. Finally, utilizing the convergence of Blaschke products to vector fields, we compute the rate at which the Weil-Petersson metric decays along radial degenerations.Comment: 49 pages, 7 figure

Sharp Spectral Asymptotics for 2-dimensional Schr\"odinger operator with a strong but degenerating magnetic field. II

I consider the same operator as in part I assuming however that $\mu \ge Ch^{-1}$ and $V$ is replaced by $(2l+1)\mu h F+W$ with l\in \bZ^+. Under some non-degeneracy conditions I recover remainder estimates up to $O \bigl(\mu^{-{\frac 1 \nu}}h^{-1} +1\bigr)$ but now case $\mu \ge Ch^{-\nu}$ is no more forbidden and the principal part is of magnitude $\mu h^{-1}$.Comment: 31 p

Quasicircles of dimension 1+k^2 do not exist

A well-known theorem of S. Smirnov states that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$. Here, we show that the precise upper bound $D(k) = 1+\Sigma^2 k^2 + \mathcal O(k^{8/3-\varepsilon})$ where $\Sigma^2$ is the maximal asymptotic variance of the Beurling transform, taken over the unit ball of $L^\infty$. The quantity $\Sigma^2$ was introduced in a joint work with K. Astala, A. Per\"al\"a and I. Prause where it was proved that $0.879 < \Sigma^2 \le 1$, while recently, H. Hedenmalm discovered that surprisingly $\Sigma^2 <1$. We deduce the asymptotic expansion of $D(k)$ from a more general statement relating the universal bounds for the integral means spectrum and the asymptotic variance of conformal maps. Our proof combines fractal approximation techniques with the classical argument of J. Becker and Ch. Pommerenke for estimating integral means.Comment: 30 page

Asymptotics of the ground state energy in the relativistic settings

The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.Comment: 19p

Short Loops and Pointwise Spectral Asymptotics

We consider pointwise semiclassical spectral asymptotics i.e. asymptotics of $e(x,x,0)$ as $h\to +0$ where $e(x,y,\tau)$ is the Schwartz kernel of the spectral projector and consider two cases when schort loops give contribution above $O(h^{1-d})$: (i) Schroedinger operator in dimensions $1,2$ as potential $V=0\implies \nabla V\ne 0$; (ii) Operators near boundaries.Comment: 23 pp, 4 fig