arXiv subjects
Publications and source records attributed to Hynek Kovařík.
We consider a critical quasilinear operator $-Δ_p u +V|u|^{p-2}u$ in $\mathbb{R}^N$ perturbed by a weakly coupled potential. For $N>p$, we find the leading asymptotic of the lowest eigenvalue of such an operator in the weak coupling limit separately for $N>p^2$ and $N\leq p^2$.
By the Aharonov-Casher theorem, the Pauli operator $P$ has no zero eigenvalue when the normalized magnetic flux $α$ satisfies $|α|<1$, but it does have a zero energy resonance. We prove that in this case a Lieb-Thirring inequality for the $γ$-th moment of the eigenvalues of $P+V$ is valid under the optimal restrictions $γ\geq |α|$ and $γ>0$. Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order $γ\geq 1$.
It is well-known that, due to the interaction between the spin and the magnetic field, the two-dimensional Pauli operator has an eigenvalue $0$ at the threshold of its essential spectrum. We show that when perturbed by an effectively positive perturbation, $V$, coupled with a small parameter $\varepsilon$, these eigenvalues become resonances. Moreover, we derive explicit expressions for the leading terms of their imaginary parts in the limit $\varepsilon\searrow 0$. These show, in particular, that the dependence of the imaginary part of the resonances on $\varepsilon$ is determined by the flux of the magnetic field. The cases of non-degenerate and degenerate zero eigenvalue are treated separately. We also discuss applications of our main results to particles with anomalous magnetic moments.
We describe the asymptotic behavior of positive solutions $u_ε$ of the equation $-Δu + au = 3\,u^{5-ε}$ in $Ω\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_ε$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Δu + (a+εV) u = 3\,u^5$ in $Ω$.
We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of these expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows to compute explicitly all the singular terms in the associated resolvent expansions. As an application we show how the magnetic field influences the time decay of the associated wave-functions quantifying thereby the paramagnetic and diamagnetic effects of the spin.
We consider the semi-classical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm-Kovařík-Portmann and Helffer-Sundqvist for the asymptotics of the ground state energy in the semi-classical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disc are discussed. Various natural conjectures are disproved and this leaves the research of an optimal result in the general case still open.
We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-Δ^D_Ω$ in locally twisted three-dimensional tubes $Ω$. In particular, we show that for any fixed $x$ the heat kernel decays for large times as $\mathrm{e}^{-E_1t}\, t^{-3/2}$, where $E_1$ is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined on a straight tube. We also discuss some applications of this result, such as Sobolev inequalities and spectral estimates for Schrödinger operators $-Δ^D_Ω-V$.
We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of "slowing down" the twisting in the mean gives rise to a non-empty discrete spectrum.