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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
core

Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators

Advances in Nonlinear Analysis
We provide bounds for the sequence of eigenvalues {λi(Ω)}i{\left\{{\lambda }_{i}\left(\Omega )\right\}}_{i} of the Dirichlet problem (I−Δ)lnu=λuinΩ,u=0inRN\Ω,{\left(I-\Delta )}^{\mathrm{ln}}u=\lambda u\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.
Chen Huyuan, Cheng Li
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Minimizing Schrödinger eigenvalues for confining potentials

Advanced Nonlinear Studies
We consider the problem of minimizing the lowest eigenvalue of the Schrödinger operator −Δ + V in L2(Rd) ${L}^{2}({\mathbb{R}}^{d})$ when the integral ∫e −tV  dx is given for some t > 0. We show that the eigenvalue is minimal for the harmonic oscillator
Frank Rupert L.
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A technique for non-intrusive greedy piecewise-rational model reduction of frequency response problems over wide frequency bands. [PDF]