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Restricted interpolation by meromorphic inner functions

100%

Poltoratski A. , Rupam R.

Concrete Operators

2016

tom 3

nr 1

102-111

Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.

A geometric estimate for a periodic Schrödinger operator

88%

Friedrich T.

Colloquium Mathematicum

2000

tom 83

nr 2

209-216

We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $-4{d^2}/{ds^2} + κ^2(s)$ with potential given by the curvature of a closed curve.

Potential theory of Schrödinger operator based on fractional Laplacian

75%

Bogdan K. , Byczkowski T.

Probability and Mathematical Statistics

2000

tom Vol. 20, Fasc. 2

293--335

We develop potential theory of Schrödinger operators based on fractional Laplacian on Euclidean spaces of arbitrary dimension. We focus on questions related to gaugeability and existence of q-harmonic functions. Results are obtained by analyzing properties of a symmetric α-stable Lévy process on Rd, including the recurrent case. We provide some relevant techniques and apply them to give explicit examples of gauge functions for a general class of domains.

Some remarks on Bochner-Riesz means

75%

Thangavelu S.

Colloquium Mathematicum

2000

tom 83

nr 2

217-230

We study $L^p$ norm convergence of Bochner-Riesz means $S_R^δ f$ associated with certain non-negative differential operators. When the kernel $S_R^m(x,y)$ satisfies a weak estimate for large values of m we prove $L^p$ norm convergence of $S_R^δ f$ for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.

The strong maximum principle for Schrödinger operators on fractals

63%

Ionescu M. V. , Okoudjou K. A. , Rogers L. G.

Demonstratio Mathematica

2019

tom Vol. 52, nr 1

404--409

We prove a strong maximum principle for Schrödinger operators defined on a class of postcritically finite fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.