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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.
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