An estimate for the number of bound states of the Schrödinger operator in two dimensions [PDF]
, 2004 For the Schrödinger operator -Δ + V on R^2 be the number of bound states. One obtains the following estimate: N(V) ≤ 1 + ∫_(R^2)∫_(R^2)|V(x)|V(y)|C_(1)ln|x-y|+C_2|^2 dx dy where C_1 = -1/2π and C_2 = (ln2-γ)/2π (γ is the Euler constant).
Stoiciu, Mihai
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Separation of Coupled Systems of Schrodinger Equations by Darboux transformations
, 2011 Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple systems of ...
Magri F. +3 more
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Smoothing estimates for variable coefficients Schroedinger equation with electromagnetic potentials
, 2012 In this paper we develop the classical multiplier technique to prove a virial identity and smoothing estimates (in a perturbative setting) for the electromagnetic variable coefficients Schroedinger equation.Comment: 15 ...
Cacciafesta, Federico
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New bounds on the Lieb-Thirring constants
, 1999 Improved estimates on the constants $L_{\gamma,d}$, for $1 ...
Hundertmark, D., Laptev, A., Weidl, T.
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A recipe for making materials with negative refraction in acoustics
, 2007 A recipe is given for making materials with negative refraction in acoustics, i.e., materials in which the group velocity is directed opposite to the phase velocity.
A.G. Ramm, Agranovich, Ramm, Ramm, Ramm
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Semiclassical analysis for a Schr\"odinger operator with a U(2) artificial gauge: the periodic case [PDF]
, 2014 We consider a Schr\"odinger operator with a Hermitian 2x2 matrix-valued potential which is lattice periodic and can be diagonalized smoothly on the whole $R^n.$ In the case of potential taking its minimum only on the lattice, we prove that the well-known
Morame, Abderemane, Truc, Francoise
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Multiple Aharonov--Bohm eigenvalues: the case of the first eigenvalue on the disk
, 2018 It is known that the first eigenvalue for Aharonov--Bohm operators with half-integer circulation in the unit disk is double if the potential's pole is located at the origin. We prove that in fact it is simple as the pole $a\neq 0$
Abatangelo, Laura
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A counterexample to an endpoint bilinear Strichartz inequality
, 2006 The endpoint Strichartz estimate $\| e^{it\Delta} f \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)}$ is known to be false by the work of Montgomery-Smith, despite being only ``logarithmically far'' from being true in some sense.
Tao, Terence
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An Agmon-Allegretto-Piepenbrink principle for Schrödinger operators. [PDF]
Rev R Acad Cienc Exactas Fis Nat A Mat, 2022 Buccheri S, Orsina L, Ponce AC.
europepmc +1 more source
On a Kirchhoff type problems with potential well and indefinite potential
, 2015 In this paper, we study the following Kirchhoff type problem:% $$ \left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\alpha,\lambda})}
Huang, Yisheng, Liu, Zeng, Wu, Yuanze
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