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 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.
We consider the twisted waveguide $Omega_theta$, i.e. the domain obtained by the rotation of the bounded cross section $omega subset {mathbb R}^{2}$ of the straight tube $Omega : = omega times {mathbb R}$ at angle $theta$ which depends on the variabl
e along the axis of $Omega$. We study the spectral properties of the Dirichlet Laplacian in $Omega_theta$, unitarily equivalent under the diffeomorphism $Omega_theta to Omega$ to the operator $H_{theta}$, self-adjoint in ${rm L}^2(Omega)$. We assume that $theta = beta - epsilon$ where $beta$ is a $2pi$-periodic function, and $epsilon$ decays at infinity. Then in the spectrum $sigma(H_beta)$ of the unperturbed operator $H_beta$ there is a semi-bounded gap $(-infty, {mathcal E}_0^+)$, and, possibly, a number of bounded open gaps $({mathcal E}_j^-, {mathcal E}_j^+)$. Since $epsilon$ decays at infinity, the essential spectra of $H_beta$ and $H_{beta - epsilon}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{beta - epsilon}$ near an arbitrary fixed spectral edge ${mathcal E}_j^pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $sigma_{rm disc}(H_{beta-epsilon})$ in a neighbourhood of ${mathcal E}_j^pm$. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of $sigma_{rm disc}(H_{beta-epsilon})$ near ${mathcal E}_j^pm$ could be represented as a finite orthogonal sum of operators of the form $-mufrac{d^2}{dx^2} - eta epsilon$, self-adjoint in ${rm L}^2({mathbb R})$; here, $mu > 0$ is a constant related to the so-called effective mass, while $eta$ is $2pi$-periodic function depending on $beta$ and $omega$.
We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinki
ng of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Further, we assume that there exists an appropriate $V$, homogeneous of order $-rho$ with $rho in (0,1)$, such that $V(x) = V(x) + O(|x|^{-rho - epsilon})$, $epsilon > 0$, as $|x| to infty$, and investigate the asymptotic distribution of the eigenvalues within a given cluster, as $q to infty$. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of $V$.
Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_ell : = H_{0, ell} - V$, $ell =D,N$, where the scalar potential $V$ is non neg
ative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of $H_D$ and $H_N$ below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of $H_ell$ near $inf sigma_{ess}(H_ell) = inf sigma(H_{0,ell})$, $ell = D,N$. Applying these Hamiltonians, we show that $sigma_{disc}(H_D)$ is infinite even if $V$ has a compact support, while $sigma_{disc}(H_N)$ could be finite or infinite depending on the decay rate of $V$.
We consider a 2D Schroedinger operator H0 with constant magnetic field, on a strip of finite width. The spectrum of H0 is absolutely continuous, and contains a discrete set of thresholds. We perturb H0 by an electric potential V which decays in a sui
table sense at infinity, and study the spectral properties of the perturbed operator H = H0 + V . First, we establish a Mourre estimate, and as a corollary prove that the singular continuous spectrum of H is empty, and any compact subset of the complement of the threshold set may contain at most a finite set of eigenvalues of H, each of them having a finite multiplicity. Next, we introduce the Krein spectral shift function (SSF) for the operator pair (H,H0). We show that this SSF is bounded on any compact subset of the complement of the threshold set, and is continuous away from the threshold set and the eigenvalues of H. The main results of the article concern the asymptotic behaviour of the SSF at the thresholds, which is described in terms of the SSF for a pair of effective Hamiltonians.
