To the Spectral Theory of the Bessel Operator on Finite Interval and Half-Line

Ananieva, Aleksandra; Budika, Viktoriya

doi:10.48550/arxiv.1603.02997

Cited by 1 publication

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“…In the paper [24] and in the preprint [3] the Friedrichs realization of the operator (1.2) is also considered for m ≥ 0. In the former paper some dispersive estimates are provided for the evolution group {e −itHm } t∈R with an emphasis in the dependence in m. In [3] a new study of the expression (1.2) on finite intervals or on the half-line is performed with the recently introduced approach of boundary triplets. The Krein and the Friedrichs extensions are indeed considered on the half-line, but the parameter m is always real.…”

Section: Introductionmentioning

confidence: 99%

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

Dereziński

Richard

2016

Ann. Henri Poincaré

135

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Abstract. The paper is devoted to operators given formally by the expressionThis expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real α, or closed operator for complex α, we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on L 2 (R+), which we denote Hm,κ and H ν 0 , with m 2 = α, −1 < Re(m) < 1, and where κ, ν ∈ C ∪{∞} specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always [0, ∞[. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that −1 < Re(m) < 1 is the maximal region of parameters for which the operators Hm,κ can be defined within the framework of the Hilbert space L 2 (R+).

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Section: Introductionmentioning

confidence: 99%