Realizations of Differential Operators on Conic Manifolds with Boundary

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L_p-Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index.Comment: 41 pages, 1 figur

Impact of KChIP2 on Cardiac Electrophysiology and the Progression of Heart Failure

Electrophysiological remodeling of cardiac potassium ion channels is important in the progression of heart failure. A reduction of the transient outward potassium current (Ito) in mammalian heart failure is consistent with a reduced expression of potassium channel interacting protein 2 (KChIP2, a KV4 subunit). Approaches have been made to investigate the role of KChIP2 in shaping cardiac Ito, including the use of transgenic KChIP2 deficient mice and viral overexpression of KChIP2. The interplay between Ito and myocardial calcium handling is pivotal in the development of heart failure, and is further strengthened by the dual role of KChIP2 as a functional subunit on both KV4 and CaV1.2. Moreover, the potential arrhythmogenic consequence of reduced Ito may contribute to the high relative incidence of sudden death in the early phases of human heart failure. With this review, we offer an overview of the insights into the physiological and pathological roles of KChIP2 and we discuss the limitations of translating the molecular basis of electrophysiological remodeling from animal models of heart failure to the clinical setting

A class of well-posed parabolic final value problems

This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax--Milgram operators in vector distribution spaces constitute the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.Comment: 16 pages. To appear in "Applied and numerical harmonic analysis"; a reference update. Conference contribution, based on arXiv:1707.02136, with some further development

Global Theory of Quantum Boundary Conditions and Topology Change

We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold $M$ with regular boundary $\Gamma=\partial M$. The space \CM of self-adjoint extensions of the covariant Laplacian on $M$ is shown to have interesting geometrical and topological properties which are related to the different topological closures of $M$. In this sense, the change of topology of $M$ is connected with the non-trivial structure of \CM. The space \CM itself can be identified with the unitary group \CU(L^2(\Gamma,\C^N)) of the Hilbert space of boundary data L^2(\Gamma,\C^N). A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, \CC_-\cap \CC_+ (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary condition reaches the Cayley submanifold \CC_-. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space \CM is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self--adjoint boundary conditions, the space \CC_- can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold \CC_- is dual of the Maslov class of \CM.Comment: 29 pages, 2 figures, harvma

Ellipticity Conditions for the Lax Operator of the KP Equations

The Lax pseudo-differential operator plays a key role in studying the general set of KP equations, although it is normally treated in a formal way, without worrying about a complete characterization of its mathematical properties. The aim of the present paper is therefore to investigate the ellipticity condition. For this purpose, after a careful evaluation of the kernel with the associated symbol, the majorization ensuring ellipticity is studied in detail. This leads to non-trivial restrictions on the admissible set of potentials in the Lax operator. When their time evolution is also considered, the ellipticity conditions turn out to involve derivatives of the logarithm of the tau-function.Comment: 21 pages, plain Te

Asymptotics of the heat equation with `exotic' boundary conditions or with time dependent coefficients

The heat trace asymptotics are discussed for operators of Laplace type with Dirichlet, Robin, spectral, D/N, and transmittal boundary conditions. The heat content asymptotics are discussed for operators with time dependent coefficients and Dirichlet or Robin boundary conditions.Comment: A talk of P.B. Gilkey at "Quantum Gravity and Spectral Geometry", Naples, July 2001, to appear in the proceedings v2: a misprint in eq. (3) correcte

Spectral triples and manifolds with boundary

We investigate manifolds with boundary in noncommutative geometry. Spectral triples associated to a symmetric differential operator and a local boundary condition are constructed. For a classical Dirac operator with a chiral boundary condition, we show that there is no tadpole.Comment: 18 pages To appear in J. Funct. Ana

The hybrid spectral problem and Robin boundary conditions

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.Comment: 24 pages. Typos and a few factors corrected. Minor comments added. Substantial Robin additions. Substantial revisio