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Register a new userCorrugated surfaces with slow modulation and quasiclassical Weierstrass representation
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Added by
Boris Konopelchenko
Publication date
2006
fields
Physics
and research's language is
English
Authors
B.G.Konopelchenko
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Quasiclassical generalized Weierstrass representation for highly corrugated surfaces with slow modulation in the three-dimensional space is proposed. Integrable deformations of such surfaces are described by the dispersionless Veselov-Novikov hierarchy.
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A new approach is proposed for study structure and properties of the total squared mean curvature $W$ of surfaces in ${bf R}^3$. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity $W$ (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of integrable flows. The $1+1$--dimensional case and, in particular, Willmore tori of revolution, are studied in details. The Willmore conjecture is proved for the mKDV--invariant Willmore tori.
The Moutard transformation for a two-dimensional Dirac operator with a complex-valued potential is constructed. It is showed that this transformation relates the potentials of Weierstrass representations of surfaces related by a composition of the inversion and a reflection with respect to an axis. It is given an analytical description of an explicit example of such a transformation which results in a creation of double points on the spectral curve of a Dirac operator with a double-periodic potential.
We derive the lateral Casimir-Polder force on a ground state atom on top of a corrugated surface, up to first order in the corrugation amplitude. Our calculation is based on the scattering approach, which takes into account nonspecular reflections and polarization mixing for electromagnetic quantum fluctuations impinging on real materials. We compare our first order exact result with two commonly used approximation methods. We show that the proximity force approximation (large corrugation wavelengths) overestimates the lateral force, while the pairwise summation approach underestimates it due to the non-additivity of dispersion forces. We argue that a frequency shift measurement for the dipolar lateral oscillations of cold atoms could provide a striking demonstration of nontrivial geometrical effects on the quantum vacuum.
We study chaotic properties of eigenstates for periodic quasi-1D waveguides with regular and random surfaces. Main attention is paid to the role of the so-called gradient scattering which is due to large gradients in the scattering walls. We demonstrate numerically and explain theoretically that the gradient scattering can be quite strong even if the amplitude of scattering profiles is very small in comparison with the width of waveguides.
A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a nontrivial line bundle. The constructed approximation is demonstrated for the Dirac magnetic monopole on the two-dimensional sphere.
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