On the stability of dense point spectrum for self‐adjoint operators

1986 ◽  
Vol 27 (1) ◽  
pp. 71-75 ◽  
Author(s):  
Lawrence E. Thomas ◽  
C. Eugene Wayne
Keyword(s):  
Point Spectrum ◽  
Dense Point ◽  
The Stability ◽  
Adjoint Operators

scholarly journals A Note on Property(gb)and Perturbations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Qingping Zeng ◽  
Huaijie Zhong
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Point Spectrum ◽  
Bounded Linear ◽  
Approximate Point Spectrum ◽  
Weyl Spectrum ◽  
Nilpotent Operators ◽  
The Stability ◽  
Quasinilpotent Operators ◽  
Study Property

An operatorT∈ℬ(X)defined on a Banach spaceXsatisfies property(gb)if the complement in the approximate point spectrumσa(T)of the upper semi-B-Weyl spectrumσSBF+-(T)coincides with the setΠ(T)of all poles of the resolvent ofT. In this paper, we continue to study property(gb)and the stability of it, for a bounded linear operatorTacting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting withT. Two counterexamples show that property(gb)in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.

scholarly journals Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity

2021 ◽  
pp. 1-29
Author(s):  
Corrado Lattanzio ◽  
Delyan Zhelyazov
Keyword(s):  
Traveling Wave ◽  
Point Spectrum ◽  
Traveling Wave Solutions ◽  
Maximum Modulus ◽  
Negative Real Part ◽  
Sufficient Condition ◽  
Winding Number ◽  
Quantum Hydrodynamics ◽  
Nonlinear Viscosity ◽  
The Stability

In this paper, we investigate spectral stability of traveling wave solutions to 1D quantum hydrodynamics system with nonlinear viscosity in the [Formula: see text], that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum stability.

On the Embedded Eigenvalues and Dense Point Spectrum of the Stark-Like Hamiltonians

1997 ◽  
Vol 183 (1) ◽  
pp. 185-200 ◽  
Author(s):  
Sergei N. Naboko ◽  
Alexander B. Pushnitski
Keyword(s):  
Point Spectrum ◽  
Dense Point ◽  
Embedded Eigenvalues

scholarly journals Stability of point spectrum for three-state quantum walks on a line

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1219-1226
Author(s):  
Martin Štefaňák ◽  
Iva Bezděková ◽  
Igor Jex ◽  
Stephen M. Barnett
Keyword(s):  
Point Spectrum ◽  
Quantum Walks ◽  
Evolution Operators ◽  
Continuous Part ◽  
Time Quantum ◽  
The Rich ◽  
The Stability ◽  
Rich Spectrum ◽  
Partial Trapping ◽  
Shed Light

Evolution operators of certain quantum walks possess, apart from the continuous part, also a point spectrum. The existence of eigenvalues and the corresponding stationary states lead to partial trapping of the walker in the vicinity of the origin. We analyze the stability of this feature for three-state quantum walks on a line subject to homogenous coin deformations. We find two classes of coin operators that preserve the point spectrum. These new classes of coins are generalizations of coins found previously by different methods and shed light on the rich spectrum of coins that can drive discrete-time quantum walks.

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