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A Two-Particle Quantum System with Zero-Range Interaction

Michele Correggi (2008/2009)

Séminaire Équations aux dérivées partielles

We study a two-particle quantum system given by a test particle interacting in three dimensions with a harmonic oscillator through a zero-range potential. We give a rigorous meaning to the Schrödinger operator associated with the system by applying the theory of quadratic forms and defining suitable families of self-adjoint operators. Finally we fully characterize the spectral properties of such operators.

A universal bound for lower Neumann eigenvalues of the Laplacian

Wei Lu, Jing Mao, Chuanxi Wu (2020)

Czechoslovak Mathematical Journal

Let M be an n -dimensional ( n 2 ) simply connected Hadamard manifold. If the radial Ricci curvature of M is bounded from below by ( n - 1 ) k ( t ) with respect to some point p M , where t = d ( · , p ) is the Riemannian distance on M to p , k ( t ) is a nonpositive continuous function on ( 0 , ) , then the first n nonzero Neumann eigenvalues of the Laplacian on the geodesic ball B ( p , l ) , with center p and radius 0 < l < , satisfy 1 μ 1 + 1 μ 2 + + 1 μ n l n + 2 ( n + 2 ) 0 l f n - 1 ( t ) d t , where f ( t ) is the solution to f ' ' ( t ) + k ( t ) f ( t ) = 0 on ( 0 , ) , f ( 0 ) = 0 , f ' ( 0 ) = 1 .

About a Pólya-Schiffer inequality

Bodo Dittmar, Maren Hantke (2011)

Annales UMCS, Mathematica

For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues λ of the fixed membrane for any n the following inequality holds [...] where λ(o) are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

About asymptotic approximations in thin waveguides

Nicole Turbe, Louis Ratier (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

About asymptotic approximations in thin waveguides

Nicole Turbe, Louis Ratier (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

Accurate Spectral Asymptotics for periodic operators

Victor Ivrii (1999)

Journées équations aux dérivées partielles

Asymptotics with sharp remainder estimates are recovered for number 𝐍 ( τ ) of eigenvalues of operator A ( x , D ) - t W ( x , x ) crossing level E as t runs from 0 to τ , τ . Here A is periodic matrix operator, matrix W is positive, periodic with respect to first copy of x and decaying as second copy of x goes to infinity, E either belongs to a spectral gap of A or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.

Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds

William Bordeaux Montrieux, Johannes Sjöstrand (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.

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