We consider the 3D Schrodinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $Ome
ga_{rm in} subset {mathbb R}^3$. We introduce the Krein spectral shift functions $xi(E;H_pm,H_0)$, $E geq 0$, for the operator pairs $(H_pm,H_0)$, and study their singularities at the Landau levels $Lambda_q : = b(2q+1)$, $q in {mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $xi(E;H_+,H_0)$ remains bounded as $E uparrow Lambda_q$, $q in {mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $xi(E;H_-,H_0)$ as $E uparrow Lambda_q$, and of $xi(E;H_pm,H_0)$ as $E downarrow Lambda_q$. The first two terms are independent of the perturbation while the third one involves the {em logarithmic capacity} of the projection of $Omega_{rm in}$ onto the plane perpendicular to $B$.
We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Lambda_q$, $q in {mathbb Z}_+$. We perturb $H_0$ by a non-local potenti
al written as a bounded pseudo-differential operator ${rm Op}^{rm w}({mathcal V})$ with real-valued Weyl symbol ${mathcal V}$, such that ${rm Op}^{rm w}({mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{{mathcal V}} = H_0 + {rm Op}^{rm w}({mathcal V})$. First, we construct symbols ${mathcal V}$, possessing a suitable symmetry, such that the operator $H_{mathcal V}$ admits an explicit eigenbasis in $L^2({mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{mathcal V}$ adjoining any given $Lambda_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${mathcal T}_q({mathcal V}) = p_q {rm Op}^{rm w}({mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${rm Ker}(H_0 - Lambda_q I)$, and investigate its spectral asymptotics.
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 di
screte eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.
We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists an almost periodic magnetic potentia
l which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} eq 0$, then generically $operatorname{dim} operatorname{Ker} H = infty$. If $b_{0} = 0$, then for each $m in {mathbb N} cup { infty }$, we construct almost periodic $b$ such that $operatorname{dim} operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.
We consider the Dirichlet Laplacian $H_gamma$ on a 3D twisted waveguide with random Anderson-type twisting $gamma$. We introduce the integrated density of states $N_gamma$ for the operator $H_gamma$, and investigate the Lifshits tails of $N_gamma$, i
.e. the asymptotic behavior of $N_gamma(E)$ as $E downarrow inf {rm supp}, dN_gamma$. In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.
We consider Schrodinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an investigation of randomly twisted waveguides.
We consider harmonic Toeplitz operators $T_V = PV:{mathcal H}(Omega) to {mathcal H}(Omega)$ where $P: L^2(Omega) to {mathcal H}(Omega)$ is the orthogonal projection onto ${mathcal H}(Omega) = left{u in L^2(Omega),|,Delta u = 0 ; mbox{in};Omegaright}$
, $Omega subset {mathbb R}^d$, $d geq 2$, is a bounded domain with $partial Omega in C^infty$, and $V: Omega to {mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V in S_{p, {rm w}}$, the weak counterpart of $S_p$. Next, we assume that $Omega$ is the unit ball in ${mathbb R}^d$, and $V = overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $partial Omega$ or $V$ is compactly supported in $Omega$. Further, we consider general $Omega$ and $V geq 0$ which is regular in $Omega$, and admits a power-like decay of rate $gamma > 0$ at $partial Omega$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-gamma$, self-adjoint in $L^2(partial Omega)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K geq 0$, self-adjoint in $L^2(Omega)$; it is known that ${rm Ker},K = {mathcal H}(Omega)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V in C(overline{Omega};{mathbb R})$, and show that $sigma_{rm ess}(K+V) = V(partial Omega)$. Assuming that $V geq 0$ and $V{|partial Omega} = 0$, we study the asymptotic distribution of the eigenvalues of $K pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.
We consider Schr{o}dinger operators on $L^{2}({mathbb R}^{d})otimes L^{2}({mathbb R}^{ell})$ of the form $ H_{omega}~=~H_{perp}otimes I_{parallel} + I_{perp} otimes {H_parallel} + V_{omega}$, where $H_{perp}$ and $H_{parallel}$ are Schr{o}dinger oper
ators on $L^{2}({mathbb R}^{d})$ and $L^{2}({mathbb R}^{ell})$ respectively, and $ V_omega(x,y)$ : = $sum_{xi in {mathbb Z}^{d}} lambda_xi(omega) v(x - xi, y)$, $x in {mathbb R}^d$, $y in {mathbb R}^ell$, is a random surface potential. We investigate the behavior of the integrated density of surface states of $H_{omega}$ near the bottom of the spectrum and near internal band edges. The main result of the current paper is that, under suitable assumptions, the behavior of the integrated density of surface states of $H_{omega}$ can be read off from the integrated density of states of a reduced Hamiltonian $H_{perp}+W_{omega}$ where $W_{omega}$ is a quantum mechanical average of $V_{omega}$ with respect to $y in {mathbb R}^ell$. We are particularly interested in cases when $H_{perp}$ is a magnetic Schr{o}dinger operator, but we also recover some of the results from [24] for non-magnetic $H_{perp}$.
We consider the Schrodinger operator $H_{eta W} = -Delta + eta W$, self-adjoint in $L^2({mathbb R}^d)$, $d geq 1$. Here $eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behav
iour of the discrete spectrum of $H_{eta W}$ near the origin, and due to the irregular decay of $eta W$, we encounter some non semiclassical phenomena. In particular, $H_{eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations with power-like decay, exponential decay or compact support.
