Mathematical Research Letters

Volume 19 (2012)

Number 2

Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian

Pages: 417 – 429



Abderemane Morame (Université de Nantes, Faculté des Sciences, Dpt. Math., UMR 6629 du CNRS, B.P. 99208, 44322 Nantes Cedex 3, France.)

Françoise Truc (Université de Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, B.P. 74, 38402 St Martin d’Hères Cedex, France.)


We consider a non-compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphicto ${\bf{X}}\,\times\, ]1,+\infty [$ with metric $ds^2=(h+dy^2)/y^{2\delta}$. {\bf{X}} is a compact manifold equipped withthe metric $h$. For a one-form $A$ on {\bf{M}} such that in each cusp $A$ is a non-exact one-form on the boundary atinfinity, we prove that the magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ satisfies the Weyl asymptotic formulawith sharp remainder. We deduce an upper bound for the counting function of the embedded eigenvalues of theLaplace–Beltrami operator $-\Delta =-\Delta_0$.

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