139 research outputs found

    Magnetic spectral bounds on starlike plane domains

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    We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that j=1nΦ(λjA/G)\sum_{j=1}^n \Phi \big( \lambda_j A/G \big) is maximal for a disk whenever Φ\Phi is concave increasing, n1n \geq 1, the domain has area AA, and λj\lambda_j is the jj-th Dirichlet eigenvalue of the magnetic Laplacian (i+β2A(x2,x1))2\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2. Here the flux β\beta is constant, and the scale invariant factor GG penalizes deviations from roundness, meaning G1G \geq 1 for all domains and G=1G=1 for disks

    On the Bohr inequality

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    The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius rr, 0<r<10<r<1, such that n=0anrn1\sum_{n=0}^\infty |a_n|r^n \leq 1 holds whenever n=0anzn1|\sum_{n=0}^\infty a_nz^n|\leq 1 in the unit disk D\mathbb{D} of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be 1/3.1/3. This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in D,\mathbb{D}, as well as for analytic functions from D\mathbb{D} into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in D.\mathbb{D}. The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the nn-dimensional Bohr radius

    Uniqueness on the Class of Odd-Dimensional Starlike Obstacles with Cross Section Data

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    We determine the uniqueness on starlike obstacles by using the cross section data. We see cross section data as spectral measure in polar coordinate at far field. Cross section scattering data suffice to give the local behavior of the wave trace. These local trace formulas contain the geometric information on the obstacle. Local wave trace behavior is connected to the cross section scattering data by Lax-Phillips' formula. Once the scattering data are identical from two different obstacles, the short time behavior of the localized wave trace is expected to give identical heat/wave invariants

    Lower bounds for the first eigenvalue of the magnetic Laplacian

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    We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.Comment: Replaces in part arXiv:1611.0193

    Lower bounds for the first eigenvalue of the magnetic Laplacian

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    We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.Comment: Replaces in part arXiv:1611.0193
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