To the spectral theory of the Bessel operator on finite interval and half-line

Ananieva, Aleksandra; Budika, Viktoriya

doi:10.1007/s10958-015-2620-1

Cited by 16 publications

(14 citation statements)

References 16 publications

(39 reference statements)

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“…In addition a sentence like "for m ≥ 1 no scattering is possible between H D and H m " (see page 85 of that paper) is in contradiction with the scattering theory developed in our Section 4 and even further extended in the subsequent sections. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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%

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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“…The fact that D(L min α ) does not depend on α for real α ∈ [0, 1[ was first proven in [1], see also [2,3]. Actually, the arguments of [1] are enough to extend the result to complex α such that |α − 1…”

Section: Comparison With Literaturementioning

confidence: 93%

On the Domains of Bessel Operators

Dereziński

Georgescu

2021

Ann. Henri Poincaré

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We consider the Schrödinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ ( m 2 - 1 4 ) 1 x 2 , often called the Bessel operator. We assume that m is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for $$|\mathrm{Re}(m)|<1$$ | Re ( m ) | < 1 and of its unique closed realization for $$\mathrm{Re}(m)>1$$ Re ( m ) > 1 coincide with the minimal second-order Sobolev space. On the other hand, if $$\mathrm{Re}(m)=1$$ Re ( m ) = 1 the minimal second-order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator. The properties of Bessel operators are compared with the properties of the corresponding bilinear forms.

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“…Therefore, self-adjoint restrictions of H max (or in other words, self-adjoint realizations of τ in L 2 (R + )) form a 1-parameter family. More precisely (see, e.g., [7] and also [1]), the following limits…”

Section: Self-adjoint Realizations and Their Spectral Propertiesmentioning

confidence: 99%

“…with the angular momentum |l| < 1 2 and self-adjoint boundary conditions at x = 0 parameterized by a parameter α ∈ [0, π) (the definition is given in Section 2, see (2.1)-(2.2) -for recent discussion of this family of operators see [1,4]). More precisely, we are interested in the dependence of the L 1 → L ∞ dispersive estimates associated to the evolution group e −itHα on the parameters α ∈ [0, π) and l ∈ (−1/2, 1/2).…”

Section: Introductionmentioning

confidence: 99%

Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions

Holzleitner

Kostenko

2016

OpMath

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Abstract.We investigate the dependence of the L 1 → L ∞ dispersive estimates for one-dimensional radial Schrödinger operators on boundary conditions at 0. In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum is positive, l ∈ (0, 1/2). However, for nonpositive angular momenta, l ∈ (−1/2, 0], the standard) decay remains true for all self-adjoint realizations.

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To the spectral theory of the Bessel operator on finite interval and half-line

Cited by 16 publications

References 16 publications

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

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

On the Domains of Bessel Operators

Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions

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