M operators: a generalisation of Weyl–Titchmarsh theory

Amrein, Werner O.; Pearson, D. B.

doi:10.1016/j.cam.2004.01.020

Cited by 70 publications

(85 citation statements)

References 8 publications

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“…[15] did not make the modification by composition with Λ ± 1 2 , but worked directly with (3.16). This was in order to avoid introducing too many operators.…”

Section: Boundary Tripletsmentioning

confidence: 99%

“…Many of the results proved for the symmetric case have subsequently been extended to this situation: see, for instance, Malamud and Mogilevski [32] for adjoint pairs of operators, and Malamud and Mogilevski [33,34] for adjoint pairs of linear relations. Amrein and Pearson [1] generalised several results from the classical Weyl-m-function for the one-dimensional SturmLiouville problem to the case of Schrödinger operators, calling them M -functions, in particular they were able to show nesting results for families of M -functions on spherical exterior domains in R 3 . For a recent contribution with applications to PDEs and characterisation of eigenvalues as poles of an operator valued Weyl-M -function, we refer the reader to [7].…”

Section: Introductionmentioning

confidence: 99%

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M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Brown

Grubb

Wood

2009

Mathematische Nachrichten

140

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Key words Closed extension, M -function, abstract boundary spaces, boundary triplets, elliptic PDEs, pseudodifferential boundary operators, essential spectrum MSC (2000) 35J25, 35J30, 35J55, 35P05, 47A10, 47A11 Dedicated to the memory of Leonid R. VolevichIn this paper, we combine results on extensions of operators with recent results on the relation between the M -function and the spectrum, to examine the spectral behaviour of boundary value problems. M -functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M -function.

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“…[15] did not make the modification by composition with Λ ± 1 2 , but worked directly with (3.16). This was in order to avoid introducing too many operators.…”

Section: Boundary Tripletsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Brown

Grubb

Wood

2009

Mathematische Nachrichten

140

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“…The systematic treatment of this topic faces certain difficulties arising from the lack of a convenient representation of Weyl-Titchmarsh functions. Indeed, neither abstract forms of Herglotz integral, nor more detailed representations found in works [4,17] for some special cases of partial differential operators, are explicit enough to serve as a foundation for the general theory. The present paper links the WeylTitchmarsh functions theory with the theory of linear systems in precise manner.…”

Section: Introduction and Notationmentioning

confidence: 99%

“…The concept of Weyl-Titchmarsh function is known for the general SturmLiouville problem, difference operators, orthogonal polynomials, Hamiltonian systems [6,10,26], various classes of extensions of symmetric operators studied within the theory of boundary and quasi-boundary triplets [7,21], and some elliptic partial differential operators [17,22,41], where it is conventionally called the Dirichlet-toNeumann map. The recent remarkable work [4] by W. O. Amrein and D. B. Pearson develops the notion of Weyl function for the three dimensional Schrödinger operator defined in the whole space. Their approach differs from mentioned above in that there is no boundary given a priori, and the authors have to introduce it artificially.…”

Section: Introduction and Notationmentioning

confidence: 99%

Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

Ryzhov

2007

Complex Anal. Oper. Theory

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The paper defines the Weyl-Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livšic. Examples of boundary value problems for partial differential equations and calculations of their Weyl-Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl-Titchmarsh function for the three dimensional Schrödinger operator introduced by W. O. Amrein and D. B. Pearson. Relationships to the Schrödinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted. Mathematics Subject Classification (2000). Primary 47A48, 47B25, 47F05; Secondary 35J10, 47N70.Keywords. Abstract boundary value problem, Green formula, Weyl-Titchmarsh function, operator colligations, linear systems with boundary control, Schrödinger operator. V. RyzhovComp.an.op.th.

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“…), we refer, for instance, to [2], [3], [4], [11], [12], [13]- [19], [24]- [28], [33]- [39], [42], [44,Ch. 3], [45], [46,Ch.…”

Section: Ac([0 R]) Denotes the Set Of Absolutely Continuous Functionmentioning

confidence: 99%

Boundary Data Maps for Schrödinger Operators on a Compact Interval

Clark

Gesztesy

Mitrea

2010

Math. Model. Nat. Phenom.

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Abstract. We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0, R] with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context.Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrödinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to different (separated) boundary conditions, and a derivation of the Herglotz property of boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the special self-adjoint case.

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M operators: a generalisation of Weyl–Titchmarsh theory

Cited by 70 publications

References 8 publications

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

Boundary Data Maps for Schrödinger Operators on a Compact Interval

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