Self-adjoint elliptic operators and manifold decompositions part I: Low Eigenmodes and stretching (original) (raw)
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Self‐adjoint elliptic operators and manifold decompositions Part II: Spectral flow and Maslov index
1996
This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous “static” results in obtaining results on the decomposition of spectral flows. Some of these “dynamic” results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of “jumping Lagrangians.” In Part 111, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce “canonical perturbations” of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson’s ...
The Maslov index and the spectra of second order elliptic operators
Advances in Mathematics
We consider second order elliptic differential operators on a bounded Lipschitz domain Ω. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in H 1 (Ω), and Lagrangian planes in H 1/2 (∂Ω) × H −1/2 (∂Ω). Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H 1/2 (∂Ω)× H −1/2 (∂Ω). Furthermore, we compute the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of the second order operators: the θ−periodic Schrödinger operators on a period cell Q ⊂ R n , the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödinger operators on starshaped domains. Our work is built on the techniques recently developed by B. Booß-Bavnbek, K. Furutani, and C. Zhu, and extends the scope of validity of their spectral flow formula by incorporating the self-adjoint extensions of the second order operators with domains in the first order Sobolev space H 1 (Ω). In addition, we generalize the results concerning relations between the Maslov and Morse indices quite recently obtained by G. Cox, J. Deng, C. Jones, J. Marzuola, A. Sukhtayev and the authors. Finally, we describe and study a link between the theory of abstract boundary triples and the Lagrangian description of self-adjoint extensions of abstract symmetric operators. Contents 1. Introduction 2 2. Self-adjoint extensions and Lagrangian planes 6 2.1. Assumptions 6 2.2. The Lagrangian planes and the self-adjoint extensions of differential operators 9 2.3. The Maslov index 13 3. The Maslov index for second order elliptic operators on smooth domains 17 3.1. Weak solutions and their traces 17 3.2. The Maslov and Morse indices 21 3.3. The spectral flow and the Maslov index 26 3.4. Spectra of elliptic operators on deformed domains and the Maslov index 26 3.5. Spectra of elliptic operators with Robin boundary conditions and the Maslov index 28 4. The Maslov index for the Schrödinger operators on Lipschitz domains 30 4.1. A general result for the Schrödinger operators 30 4.2. Spectra of θ−periodic Schrödinger operators and the Maslov index 33 4.3. Spectra of Schrödinger operators on star-shaped domains 36
Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds
Pseudo-Differential Operators, Generalized Functions and Asymptotics, 2013
In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader's convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.
1 an Approach to Spectral Problems on Riemannian Manifolds
2016
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In particular, a bounded set of eigenvalues can be determined using a space of such polyharmonic functions with a fixed set of singularities. It also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polyharmonic functions with appropriate fixed set of singularities.
Isospectral Hamiltonian flows in finite and infinite dimensions
Communications in Mathematical …, 1990
The approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad*-invariant Poisson submanifolds of the dual (gl(r) +)* of the positive part of the loop algebra gl(r) is obtained through a generalization of the standard method of linearization on the Jacobi variety of the invariant spectral curve 5. These curves are embedded in the total space of a line bundle Γ-^IP^C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map J r :M Nr x M N ,.->(#/(r) +)* from the space of rank-r deformations of a fixed N x N matrix A. It is shown^how the linear flow of line bundles E t-*S over a suitably desingularized curve S may be used to determine both the flow of matricial polynomials L(λ) and the Hamiltonian flow in the space M Ntf x M NjΓ in terms of 0-functions. The resulting flows are proved to be completely integrable. The reductions to subalgebras developed in part I are shown to correspond to in variance of the spectral curves and line bundles E t-+S under certain linear or anti-linear involutions. The integration of two examples from part I is given to illustrate the method: the Rosochatius system, and the CNLS (coupled non-linear Schrόdinger) equation.
An approach to spectral problems on Riemannian manifolds
2014
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In particular, a bounded set of eigenvalues can be determined using a space of such polyharmonic functions with a fixed set of singularities. It also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polyharmonic functions with appropriate fixed set of singularities.
Annales Henri Poincaré, 2012
Problems related to symmetries and dimensional reduction are common in the mathematical and physical literature, and are intensively studied presently. As a rule, the symmetry group ("reducing group") and its orbits ("external dimensions") are compact, and this is essential in models where the volume of the orbits is related to physical quantities. However, this case is only a part of the natural problems related to dimensional reduction. In the present paper, we consider an action of a (generally non-compact) Lie group on a vector bundle, construct a formalism of reduced bundles for description of all invariant sections of the original bundle, and study the algebraic structures that occur in the reduced bundle. We show that in the case of a non-compact reducing group it is possible that the reduction is non-standard ("non-canonical"), and construct an explicit obstruction for canonical reduction in terms of cohomology of groups. We consider in detail the reduction of tangent and cotangent bundles, and show that, in general, the duality between the two is violated in the process of reduction. The reduction of the tensor product of tangent and cotangent bundles is also discussed. We construct examples of non-canonical dimensional reduction and of violation of duality between the tangent and cotangent bundles in the reduction.