Displaying similar documents to “A singular spectral identity and inequality involving the Dirichlet integral of an ordinary differential expression”

Modulation invariant and multilinear singular integral operators

Michael Christ (2005-2006)

Séminaire Bourbaki

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In a series of papers beginning in the late 1990s, Michael Lacey and Christoph Thiele have resolved a longstanding conjecture of Calderón regarding certain very singular integral operators, given a transparent proof of Carleson’s theorem on the almost everywhere convergence of Fourier series, and initiated a slew of further developments. The hallmarks of these problems are multilinearity as opposed to mere linearity, and especially modulation symmetry. By modulation is meant multiplication...

On discreteness of spectrum of a functional differential operator

Sergey Labovskiy, Mário Frengue Getimane (2014)

Mathematica Bohemica

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We study conditions of discreteness of spectrum of the functional-differential operator ℒ u = - u ' ' + p ( x ) u ( x ) + ∫ - ∞ ∞ ( u ( x ) - u ( s ) ) d s r ( x , s ) on ( - ∞ , ∞ ) . In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.

The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix

F. Štampach, P. Šťovíček (2014)

Special Matrices

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A family T(ν), ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in ℓ2(ℤ+) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for |ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic...