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We consider the self-adjoint Schrodinger operator in $L^2(mathbb{R}^d)$, $dgeq 2$, with a $delta$-potential supported on a hyperplane $Sigmasubseteqmathbb{R}^d$ of strength $alpha=alpha_0+alpha_1$, where $alpha_0inmathbb{R}$ is a constant and $alpha_1in L^p(Sigma)$ is a nonnegative function. As the main result, we prove that the lowest spectral point of this operator is not smaller than that of the same operator with potential strength $alpha_0+alpha_1^*$, where $alpha_1^*$ is the symmetric decreasing rearrangement of $alpha_1$. The proof relies on the Birman-Schwinger principle and the reduction to an analogue of the P{o}lya-SzegH{o} inequality for the relativistic kinetic energy in $mathbb{R}^{d-1}$.
We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furt
In this note the two dimensional Dirac operator $A_eta$ with an electrostatic $delta$-shell interaction of strength $etainmathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interacti
We consider the discrete spectrum of the two-dimensional Hamiltonian $H=H_0+V$, where $H_0$ is a Schrodinger operator with a non-constant magnetic field $B$ that depends only on one of the spatial variables, and $V$ is an electric potential that deca
In this note the three dimensional Dirac operator $A_m$ with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $A_m$ is self-adjoint in $L^2(Omega;mathbb{C}^4)$ for any op
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrodinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).