Stability of the essential spectrum of second-order complex elliptic operators

1998 ◽  
Vol 1998 (500) ◽  
pp. 113-126
Author(s):  
Peter Stollmann ◽  
El Maati Ouhabaz
Keyword(s):  
Essential Spectrum ◽  
Elliptic Operators ◽  
Second Order ◽  
Order Complex

scholarly journals Heat Kernels of Second Order Complex Elliptic Operators and Applications

1998 ◽  
Vol 152 (1) ◽  
pp. 22-73 ◽  
Author(s):  
Pascal Auscher ◽  
Alan McIntosh ◽  
Philippe Tchamitchian
Keyword(s):  
Elliptic Operators ◽  
Heat Kernels ◽  
Second Order ◽  
Order Complex

On the spectra of non-self-adjoint realisations of second-order elliptic operators

1981 ◽  
Vol 90 (1-2) ◽  
pp. 71-105 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Spherical Shell ◽  
Essential Spectrum ◽  
Weighted Space ◽  
Elliptic Operators ◽  
Second Order ◽  
Sectorial Operator ◽  
Image Position ◽  
Second Order Elliptic Operators ◽  
Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

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

1979 ◽  
Vol 31 (2) ◽  
pp. 419-426 ◽  
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.

18.—Asymptotic Estimates for the Lengths of the Gaps in the Essential Spectrum of Self-adjoint Differential Operators

1976 ◽  
Vol 74 ◽  
pp. 239-252 ◽  
Author(s):  
M. S. P. Eastham ◽  
W. N. Everitt
Keyword(s):  
Essential Spectrum ◽  
Differential Operators ◽  
Second Order ◽  
Asymptotic Estimates ◽  
Image Position ◽  
Differentiability Properties

SynopsisThe paper gives asymptotic estimates of the formas λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.

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