Lower bounds of the essential spectrum of the Laplace-Beltrami operator and its application to complex geometry

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).

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Lower Bounds on the Number of Points in the Lower Spectrum of Elliptic Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-045-0 ◽

1979 ◽

Vol 31 (2) ◽

pp. 419-426 ◽

Cited By ~ 2

Author(s):

Walter Allegretto

Keyword(s):

Essential Spectrum ◽

Lower Bounds ◽

Unbounded Domain ◽

Bound States ◽

Adjoint Operator ◽

Closed Surface ◽

Elliptic Operators ◽

Second Order ◽

Regular Boundary ◽

Lower Spectrum

Let G denote an unbounded domain of Euclidean m-space Em with regular boundary, and let L be a self-adjoint operator generated in L2(G) by a second order elliptic expression. We denote by S(L) the spectrum of L, by µ the least point of the essential spectrum Se(L) and by N(L) the number of bound states of L; that is, the number of points in (–∞, µ) ∩ S(L). There are many results in the literature dealing with the localization, significance and properties of µ, of Se(L) and of (–∞, µ)⌒ S(L), with most of the emphasis on the cases where G = Em or G is the exterior of a closed surface in Em. We refer the reader to the books by Glazman [12], Schechter [19], Reed and Simon [18], and Paris [9], where extensive references are also found.

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Eigenvalue estimates for submanifolds of warped product spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004113000443 ◽

2013 ◽

Vol 156 (1) ◽

pp. 25-42 ◽

Cited By ~ 2

Author(s):

G. P. BESSA ◽

S. C. GARCÍA–MARTÍNEZ ◽

L. MARI ◽

H. F. RAMIREZ–OSPINA

Keyword(s):

Essential Spectrum ◽

Lower Bounds ◽

Warped Product ◽

Minimal Submanifolds ◽

Hyperbolic Spaces ◽

Fundamental Tone ◽

Product Spaces ◽

Eigenvalue Estimates ◽

Hyperbolic Graphs ◽

Warped Product Spaces

AbstractIn this paper, we give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type Nn ×f Qq, where f ∈ C∞(N). This setting allows us to deal, among other things, with minimal submanifolds bounded by cylinders, cones, spheres and pseudo-hyperbolic spaces where most of these examples are not covered in the literature. Applications also include the study of the essential spectrum of hyperbolic graphs over compact regions of the boundary at infinity.

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