Heat kernel estimates for two-dimensional relativistic Hamiltonians with magnetic field

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.

Absence of positive eigenvalues of magnetic Schrodinger operators

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrodinger operators in $mathbb{R}^d,, dgeq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.

Energy asymptotics in the Brezis-Nirenberg problem. The higher-dimensional case

For dimensions $N geq 4$, we consider the Brezis-Nirenberg variational problem of finding [ S(epsilon V) := inf_{0 otequiv uin H^1_0(Omega)} frac{int_Omega | abla u|^2 , dx +epsilon int_Omega V, |u|^2 , dx}{left(int_Omega |u|^q , dx right)^{2/q}}, ] where $q=frac{2N}{N-2}$ is the critical Sobolev exponent and $Omega subset mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(epsilon V)$ to leading order as $epsilon to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case $N = 3$.

Energy asymptotics in the three-dimensional Brezis--Nirenberg problem

For a bounded open set $Omegasubsetmathbb R^3$ we consider the minimization problem $$ S(a+epsilon V) = inf_{0 otequiv uin H^1_0(Omega)} frac{int_Omega (| abla u|^2+ (a+epsilon V) |u|^2),dx}{(int_Omega u^6,dx)^{1/3}} $$ involving the critical Sobolev exponent. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on $a$ and $V$ we compute the asymptotics of $S(a+epsilon V)-S$ as $epsilonto 0+$, where $S$ is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to $a$ and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $S(a+epsilon V)<S$ for all sufficiently small $epsilon>0$.

On minimal decay at infinity of Hardy-weights

We study the behaviour of Hardy-weights for a class of variational quasi-linear elliptic operators of $p$-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their integrability with respect to certain integral weights. Some of the results are extended also to nonsymmetric linear elliptic operators. Applications to various examples are discussed as well.

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

Impurity-bound excitons in one and two dimensions

We study three-body Schrodinger operators in one and two dimensions modelling an exciton interacting with a charged impurity. We consider certain classes of multiplicative interaction potentials proposed in the physics literature. We show that if the impurity charge is larger than some critical value, then three-body bound states cannot exist. Our spectral results are confirmed by variational numerical computations based on projecting on a finite dimensional subspace generated by a Gaussian basis.

Robin eigenvalues on domains with peaks

Let $Omegasubsetmathbb{R}^N$, $Nge 2,$ be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian $umapsto -Delta u$ in $Omega$ with the Robin boundary condition $partial_n u=alpha u$ on $partialOmega$ with $partial_n$ being the outward normal derivative and $alpha>0$ being a parameter. We show that for large $alpha$ the associated eigenvalues $E_j(alpha)$ behave as $E_j(alpha)sim -epsilon_j alpha^ u$, where $ u>2$ and $epsilon_j>0$ depend on the dimension and the peak geometry. This is in contrast with the well-known estimate $E_j(alpha)=O(alpha^2)$ for the Lipschitz domains.

On the existence of impurity bound excitons in one-dimensional systems with zero range interactions

We consider a three-body one-dimensional Schrodinger operator with zero range potentials, which models a positive impurity with charge $kappa > 0$ interacting with an exciton. We study the existence of discrete eigenvalues as $kappa$ is varied. On one hand, we show that for sufficiently small $kappa$ there exists a unique bound state whose binding energy behaves like $kappa^4$, and we explicitly compute its leading coefficient. On the other hand, if $kappa$ is larger than some critical value then the system has no bound states.

Schroedinger operators on exterior domains with Robin boundary conditions: heat kernel estimates

We study Schroedinger operators with Robin boundary conditions on exterior domains in $R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.