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Register a new userDirichlet and Neumann Eigenvalues for Half-Plane Magnetic Hamiltonians
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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 negative, 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 $V$.
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In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $pto 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
In the presence of the homogeneous electric field ${bf E}$ and the homogeneous perpendicular magnetic field ${bf B}$, the classical trajectory of a quantum particle on ${mathbb R}^2$ moves with drift velocity $alpha$ which is perpendicular to the electric and magnetic fields. For such Hamiltonians the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper one proves this conjecture for the perturbations $V(x, y)$ which have sufficiently small support in direction of drift velocity.
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 $Omega_{rm in} subset {mathbb R}^3$. 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 $B$.
This note aims to give prominence to some new results on the absence and localization of eigenvalues for the Dirac and Klein-Gordon operators, starting from known resolvent estimates already established in the literature combined with the renowned Birman-Schwinger principle.
We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schrodinger-type operator on the boundary of the domain with boundary conditions at the corners.
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