scholarly journals On holomorphic families of Schrödinger-type operators with singular potentials on manifolds of bounded geometry

2004 ◽  
Vol 21 (3) ◽  
pp. 361-377
Author(s):  
Ognjen Milatovic
Keyword(s):  
Singular Potentials ◽  
Bounded Geometry ◽  
Schrödinger Type Operators

scholarly journals Onm-accretive Schrödinger-type operators with singular potentials on manifolds of bounded geometry

2003 ◽  
Vol 2003 (38) ◽  
pp. 2415-2423 ◽  
Author(s):  
Ognjen Milatovic
Keyword(s):  
Laplacian Operator ◽  
Sufficient Condition ◽  
Singular Potentials ◽  
Hermitian Vector Bundle ◽  
Bounded Geometry ◽  
Bochner Laplacian ◽  
Positivity Of Solution ◽  
Hermitian Connection ◽  
Schrödinger Type Operators ◽  
Kato’S Inequality

We consider a Schrödinger-type differential expression∇∗ ∇+V, where∇is aC∞-bounded Hermitian connection on a Hermitian vector bundleEof bounded geometry over a manifold of bounded geometry(M,g)with positiveC∞-bounded measuredμ, andVis a locally integrable linear bundle endomorphism. We define a realization of∇∗ ∇+VinL2(E)and give a sufficient condition for itsm-accretiveness. The proof essentially follows the scheme of T. Kato, but it requires the use of a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of solution to a certain differential equation onM.

Minimal Operators for Schrodinger-Type Differential Expressions with Discontinuous Principal Coefficients

1980 ◽  
Vol 32 (6) ◽  
pp. 1423-1437 ◽  
Author(s):  
M. Faierman ◽  
I. Knowles
Keyword(s):  
Basic Problem ◽  
Adjoint Operator ◽  
General Setting ◽  
Equivalence Classes ◽  
The Self ◽  
Singular Potentials ◽  
Image Position ◽  
Schrödinger Type Operators ◽  
Complex Valued

The objective of this paper is to extend the recent results [7, 8, 9] concerning the self-adjointness of Schrödinger-type operators with singular potentials to a more general setting. We shall be concerned here with formally symmetric elliptic differential expressions of the form1.1where x = (x1, …, xm) ∈ Rm (and m ≧ 1), i = (–1)1/2, ∂j = ∂/∂xj, and the coefficients ajk, bj and q are real-valued and measurable on Rm.The basic problem that we consider is that of deciding whether or not the formal operator defined by (1.1) determines a unique self-adjoint operator in the space L2(Rm) of (equivalence classes of) square integrable complex-valued functions on Rm.

scholarly journals Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang
Keyword(s):  
Finite Rank ◽  
Intrinsic Geometry ◽  
Bounded Geometry ◽  
Higher Dimensional ◽  
Median Algebra ◽  
Asymptotic Geometry ◽  
Median Operator

AbstractThis paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

scholarly journals Approximating pointwise products of quasimodes

2020 ◽  
Vol 32 (3) ◽  
pp. 541-552
Author(s):  
Mei Ling Jin
Keyword(s):  
Vector Space ◽  
Harmonic Analysis ◽  
Riemannian Manifolds ◽  
Spectral Projection ◽  
Beltrami Operator ◽  
Singular Potentials ◽  
Link Type ◽  
Right Hand ◽  
Low Degree ◽  
The Right

AbstractWe obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space {B_{n}}, and we prove that the size of the space {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions {d=2,3}, {d=4,5} and {d\geq 6}, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case {\lambda=\mu} of bilinear quasimode estimates improves {L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on L^{p}-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when {d\geq 8}. And on this basis, we give approximation bounds in {H^{-1}}-norm. We also prove approximation bounds for the products of quasimodes in {L^{2}}-norm using the results of {L^{p}}-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.

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