On the η-invariant of certain nonlocal boundary value problems

Brüning, Jochen; Lesch, Matthias

doi:10.1215/s0012-7094-99-09613-8

Cited by 63 publications

(97 citation statements)

References 27 publications

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“…cf (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). We refer to Nicolaescu [26] for the motivation of the choice of the sign in (3-13).…”

Section: The -Adjoint Mapmentioning

confidence: 99%

“…Let c j ; a j ; h j (j D 0; : : : ; d ) be as in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12). For each j D 0; : : : ; d , set…”

Section: The Dual Complexunclassified

“…In addition, note that if we set z C D C in (2-29), then the right hand side of (2-29) is equal to P d j D0 G j . Hence, (2-29) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) imply that (3-29) is equivalent to N…”

Section: The Dual Complexmentioning

confidence: 99%

“…Using (3-8), we obtain from (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), that, for j D 0; : : : ; r 1,…”

Section: The Refined Torsion Of the Dual Complexmentioning

confidence: 99%

“…Then, by (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12), the elements h j 2 k, which appear in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), are given by…”

Section: Calculation Of the Refined Torsion In Case B Is Bijectivementioning

confidence: 99%

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Refined analytic torsion as an element of the determinant line

2007

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We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold M with coefficients in a flat complex vector bundle E . We compute the Ray-Singer norm of the refined analytic torsion. In particular, if there exists a flat Hermitian metric on E , we show that this norm is equal to 1. We prove a duality theorem, establishing a relationship between the refined analytic torsions corresponding to a flat connection and its dual.

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“…cf (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). We refer to Nicolaescu [26] for the motivation of the choice of the sign in (3-13).…”

Section: The -Adjoint Mapmentioning

confidence: 99%

“…Let c j ; a j ; h j (j D 0; : : : ; d ) be as in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12). For each j D 0; : : : ; d , set…”

Section: The Dual Complexunclassified

Section: The Dual Complexmentioning

confidence: 99%

“…Using (3-8), we obtain from (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), that, for j D 0; : : : ; r 1,…”

Section: The Refined Torsion Of the Dual Complexmentioning

confidence: 99%

“…Then, by (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12), the elements h j 2 k, which appear in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), are given by…”

Section: Calculation Of the Refined Torsion In Case B Is Bijectivementioning

confidence: 99%

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Refined analytic torsion as an element of the determinant line

2007

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An Algebra of Boundary Value Problems Not Requiring Shapiro–Lopatinskij Conditions

Schulze

2001

Journal of Functional Analysis

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We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces. Academic PressKey Words: boundary value problems with APS conditions; pseudo-differential operators on manifolds with boundary; ellipticity and parametrix constructions.

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On Boundary Value Problems for Dirac Type Operators

Brüning¹,

Lesch²

2001

Journal of Functional Analysis

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In a series of papers, we will develop systematically the basic spectral theory of (self-adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self-adjoint) boundary conditions with optimal regularity, for which we will derive the heat asymptotics and index theorems in subsequent publications. Along with a number of new results, we -extend and simplify the proofs of many known theorems. Our point of departure is the simple structure which is displayed by Dirac type operators near the boundary. Thus our proofs are given in an abstract functional analytic setting, generalizing considerably the framework of compact manifolds with boundary. The results of this paper have been announced previously by the authors (J. Bru ning and M. Contents.1. Introduction. 2. Sobolev spaces and operator algebras associated with self-adjoint operators. 3. Fredholm pairs. 4. Regularity for the model operator. 5. Criteria for regularity. 6. Variable coefficients and second order operators. 7. Well-posed boundary value problems.

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On the η-invariant of certain nonlocal boundary value problems

Cited by 63 publications

References 27 publications

Refined analytic torsion as an element of the determinant line

Refined analytic torsion as an element of the determinant line

An Algebra of Boundary Value Problems Not Requiring Shapiro–Lopatinskij Conditions

On Boundary Value Problems for Dirac Type Operators

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