On Schroedinger operators with multisingular inverse-square anisotropic potentials

Felli, Veronica; Marchini, Elsa M.; Terracini, Susanna

doi:10.1512/iumj.2009.58.3471

Cited by 23 publications

(34 citation statements)

References 31 publications

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“…Combining (4.18) and (4.19) one concludes 20) and hence 22) are linearly independent (mod dom(T )), implying…”

Section: Decoupling Of Deficiency Indices An Abstract Approachmentioning

confidence: 91%

“…E\ supp (f ) = {x ∈ E | there exists r > 0 so that B n (x, r) ⊆ E and f = 0 in B n (x, r)} = E\{x ∈ E | f (x) = 0}, (A. 22) as was to be shown.…”

Section: Applications To Schrödinger-type and Second-order Elliptic Pmentioning

confidence: 93%

“…[21], [22], [26]), or a van der Waals-type potential of the form, 8) for fixed A j ∈ (0, ∞), B j ∈ R (although, physics requires B j ∈ (0, ∞)), and m 10, with m = 12 giving rise to a Lennard-Jones-type potential (cf., e.g., [47, pp. 53-59], [54,Sect.…”

Section: )mentioning

confidence: 99%

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Decoupling of deficiency indices and applications to Schrödinger-type operators with possibly strongly singular potentials

Gesztesy

Mitrea

Nenciu

2016

Advances in Mathematics

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Abstract. We investigate closed, symmetric L 2 (R n )-realizations H of Schrö-dinger-type operators (−∆ + V ) ↾ C ∞ 0 (R n \Σ) whose potential coefficient V has a countable number of well-separated singularities on compact sets Σ j , j ∈ J, of n-dimensional Lebesgue measure zero, with J ⊆ N an index set and Σ = j∈J Σ j . We show that the defect, def(H), of H can be computed in terms of the individual defects, def(with potential coefficient V j localized around the singularity Σ j , j ∈ J, where V = j∈J V j . In particular, we proveincluding the possibility that one, and hence both sides equal ∞. We first develop an abstract approach to the question of decoupling of deficiency indices and then apply it to the concrete case of Schrödinger-type operators in L 2 (R n ).Moreover, we also show how operator (and form) bounds for V relative to H 0 = −∆ ↾ H 2 (R n ) can be estimated in terms of the operator (and form) bounds of V j , j ∈ J, relative to H 0 . Again, we first prove an abstract result and then show its applicability to Schrödinger-type operators in L 2 (R n ).Extensions to second-order (locally uniformly) elliptic differential operators on R n with a possibly strongly singular potential coefficient are treated as well.

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“…Combining (4.18) and (4.19) one concludes 20) and hence 22) are linearly independent (mod dom(T )), implying…”

Section: Decoupling Of Deficiency Indices An Abstract Approachmentioning

confidence: 91%

“…E\ supp (f ) = {x ∈ E | there exists r > 0 so that B n (x, r) ⊆ E and f = 0 in B n (x, r)} = E\{x ∈ E | f (x) = 0}, (A. 22) as was to be shown.…”

Section: Applications To Schrödinger-type and Second-order Elliptic Pmentioning

confidence: 93%

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Decoupling of deficiency indices and applications to Schrödinger-type operators with possibly strongly singular potentials

Gesztesy

Mitrea

Nenciu

2016

Advances in Mathematics

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“…This class of potentials appears in a number of physical phenomena (see e.g. [15], [16], [22], [30], [32], [35] and references therein) and can be classified according to the number of singularities (poles), σ-degree of the singularity (order of the poles), dependence on directions (anisotropy) and decay at infinity. One of the most difficult cases is the one of anisotropic critical potentials, namely…”

mentioning

confidence: 99%

“…Examples of (1.3) are 4) where x i = (x i 1 , x i 2 , ..., x i n ) and d i ∈ R n are constant vectors. In the theory of Schrodinger operators, the potentials in (1.4) are called multipolar Hardy potentials and multiple dipoletype potentials, respectively (see [15] and [16]). …”

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confidence: 99%

On the Heat Equation with Nonlinearity and Singular Anisotropic Potential on the Boundary

Almeida

Ferreira

Precioso³

2016

Potential Anal

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This paper concerns with the heat equation in the half-space R n + with nonlinearity and singular potential on the boundary ∂R n + . We develop a well-posedness theory (without using Kato and Hardy inequalities) that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak L p -spaces in which we prove key linear estimates for some boundary operators arising from the Duhamel integral formulation in R n + . Moreover, we investigate qualitative properties of solutions like self-similarity, positivity and symmetry around the axis − − → Ox n .

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A Fourier Analysis Approach to Elliptic Equations with Critical Potentials and Nonlinear Derivative Terms

Ferreira

Castañeda-Centurión

2017

Milan J. Math.

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On Schroedinger operators with multisingular inverse-square anisotropic potentials

Abstract: We study positivity, localization of binding and essential self-adjointness properties of a class of Schrödinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials.

Cited by 23 publications

References 31 publications

Decoupling of deficiency indices and applications to Schrödinger-type operators with possibly strongly singular potentials

Decoupling of deficiency indices and applications to Schrödinger-type operators with possibly strongly singular potentials

On the Heat Equation with Nonlinearity and Singular Anisotropic Potential on the Boundary

A Fourier Analysis Approach to Elliptic Equations with Critical Potentials and Nonlinear Derivative Terms

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