scholarly journals Sobolev extension operators and Neumann eigenvalues

2020 ◽  
Vol 10 (1) ◽  
pp. 337-353
Author(s):  
Vladimir Gol'dshtein ◽  
Valerii Pchelintsev ◽  
Alexander Ukhlov
Keyword(s):  
Neumann Eigenvalues

Ambarzumyan-type theorem for the impulsive Sturm–Liouville operator

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ran Zhang ◽  
Chuan-Fu Yang
Keyword(s):  
Liouville Operator ◽  
Type Theorem ◽  
Sturm Liouville ◽  
Neumann Laplacian ◽  
Neumann Eigenvalues

AbstractWe prove that if the Neumann eigenvalues of the impulsive Sturm–Liouville operator {-D^{2}+q} in {L^{2}(0,\pi)} coincide with those of the Neumann Laplacian, then {q=0}.

scholarly journals Space quasiconformal composition operators with applications to Neumann eigenvalues

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Vladimir Gol’dshtein ◽  
Ritva Hurri-Syrjänen ◽  
Valerii Pchelintsev ◽  
Alexander Ukhlov
Keyword(s):  
Composition Operators ◽  
Neumann Eigenvalues

scholarly journals Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians

2014 ◽  
Vol 26 (02) ◽  
pp. 1450003 ◽  
Author(s):  
Vincent Bruneau ◽  
Pablo Miranda ◽  
Georgi Raikov
Keyword(s):  
Magnetic Field ◽  
Asymptotic Behavior ◽  
Half Plane ◽  
Compact Support ◽  
Constant Magnetic Field ◽  
Scalar Potential ◽  
Neumann Boundary ◽  
Essential Spectra ◽  
Neumann Eigenvalues ◽  
Effective Hamiltonians

Let H0,D (respectively, H0,N) be the Schrödinger operator in constant magnetic field on the half-plane with Dirichlet (respectively, Neumann) boundary conditions, and let Hℓ := H0,ℓ - V, ℓ = 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 HD and HN below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behavior of the discrete spectrum of Hℓ near inf σ ess (Hℓ) = inf σ(H0,ℓ), ℓ = D, N. Applying these Hamiltonians, we show that σ disc (HD) is infinite even if V has a compact support, while σ disc (HN) could be finite or infinite depending on the decay rate of V.

A universal bound for lower Neumann eigenvalues of the Laplacian

2019 ◽  
Vol 70 (2) ◽  
pp. 473-482
Author(s):  
Wei Lu ◽  
Jing Mao ◽  
Chuanxi Wu
Keyword(s):  
Universal Bound ◽  
Eigenvalues Of The Laplacian ◽  
Neumann Eigenvalues

scholarly journals Eigenmode analysis of the electromagnetic field scattered by an elliptic cone

2011 ◽  
Vol 9 ◽  
pp. 31-37 ◽  
Author(s):  
M. Kijowski ◽  
L. Klinkenbusch
Keyword(s):  
Free Space ◽  
Plane Electromagnetic Wave ◽  
Multipole Expansion ◽  
Neumann Condition ◽  
Electrically Conducting ◽  
Eigenmode Analysis ◽  
Elliptic Cone ◽  
Series Transformation ◽  
Neumann Eigenvalues

Abstract. The vector spherical-multipole analysis is applied to determine the scattering of a plane electromagnetic wave by a perfectly electrically conducting (PEC) semi-infinite elliptic cone. From the eigenfunction expansion of the total field in the space outside the elliptic cone, the scattered far field is obtained as a multipole expansion of the free-space type by a single integration over the induced surface currents. As for the evaluation of the free-space-type expansion it is necessary to apply suitable series transformation techniques, a sufficient number of eigenfunctions has to be considered. The eigenvalues of the underlying two-parametric eigenvalue problem with two coupled Lamé equations belong to the Dirichlet- or the Neumann condition and can be arranged as so-called eigenvalue curves. It has been observed that the eigenvalues are in two different domains: In the first one Dirichlet- and Neumann eigenvalues are either nearly coinciding, while in the second one they are strictly separated. The eigenfunctions of the first (coinciding) type look very similar to free-space modes and do not contribute to the scattered field. This observation allows to significantly improve the determination of diffraction coefficients.

EVOLUTION OF LIOUVILLE DENSITY IN BILLIARDS: THE QUANTUM CONNECTION

2006 ◽  
Vol 20 (14) ◽  
pp. 795-813
Author(s):  
DEBABRATA BISWAS
Keyword(s):  
Configuration Space ◽  
Chaotic Systems ◽  
Evolution Operator ◽  
Eigenvalues And Eigenfunctions ◽  
Exact Quantum ◽  
Classical Trajectories ◽  
Approximate Nature ◽  
One To One ◽  
Neumann Eigenvalues

The study of classical Liouville density arises naturally in chaotic systems where a probabilistic treatment is more appropriate. In this review, we show that the evolution of a density projected onto the configuration space has a quantum connection in billiard systems. The eigenvalues and eigenfunctions of the concerned evolution operator have an approximate one-to-one correspondence with the quantum Neumann eigenvalues and eigenfunctions. For exceptional billiard shapes such as the rectangle, this correspondence is even exact. Despite the approximate nature of the correspondence, we demonstrate that the exact quantum Neumann eigenstates can be used to expand and evolve an arbitrary classical projected-density. For the rectangular and stadium billiards, results are compared with the actual evolution of the density using classical trajectories and are found to be satisfactory.

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