We study semigroups generated by two-dimensional relativistic Hamiltonians with magnetic field. In particular, for compactly supported radial magnetic field we show how the long time behaviour of the associated heat kernel depends on the flux of the field. Similar questions are addressed for Aharonov-Bohm type magnetic field.
We study the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.
Using Moutard transformations we show how explicit examples of two-dimensional Schroedinger operators with fast decaying potential and multidimensional $L_2$-kernel may be constructed
We get a generalization of Kreins formula -which relates the resolvents of different selfadjoint extensions of a differential operator with regular coefficients- to the non-regular case $A=-partial_x^2+( u^2-1/4)/x^2+V(x)$, where $0< u<1$ and $V(x)$ is an analytic function of $xinmathbb{R}^+$ bounded from below. We show that the trace of the heat-kernel $e^{-tA}$ admits a non-standard small-t asymptotic expansion which contains, in general, integer powers of $t^ u$. In particular, these powers are present for those selfadjoint extensions of $A$ which are characterized by boundary conditions that break the local formal scale invariance at the singularity.
We consider a non-relativistic electron interacting with a classical magnetic field pointing along the $x_3$-axis and with a quantized electromagnetic field. The system is translation invariant in the $x_3$-direction and we consider the reduced Hamiltonian $H(P_3)$ associated with the total momentum $P_3$ along the $x_3$-axis. For a fixed momentum $P_3$ sufficiently small, we prove that $H(P_3)$ has a ground state in the Fock representation if and only if $E(P_3)=0$, where $P_3 mapsto E(P_3)$ is the derivative of the map $P_3 mapsto E(P_3) = inf sigma (H(P_3))$. If $E(P_3) eq 0$, we obtain the existence of a ground state in a non-Fock representation. This result holds for sufficiently small values of the coupling constant.