## E B: Spectral instability for some Schrödinger operators

Venue: | Numer. Math |

Citations: | 14 - 2 self |

### BibTeX

@ARTICLE{Aslanyan_eb:,

author = {A. Aslanyan and E. B. Davies},

title = {E B: Spectral instability for some Schrödinger operators},

journal = {Numer. Math},

year = {},

pages = {525--552}

}

### OpenURL

### Abstract

Abstract. We define the concept of instability index of an isolated eigenvalue of a non-self-adjoint operator, and prove some of its general properties. We also describe a stable procedure for computing this index for Schrödinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential.

### Citations

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(Show Context)
Citation Context ...) is defined to be the coefficient of s in the expansion of the eigenvalue of the perturbed operator H + sV : λ(s) = λ + a1(V )s + a2(V )s 2 + · · · Proof By standard arguments in perturbation theory =-=[14]-=- we have (H + sV )(f + sg + . . .) = (λ + sµ + . . .)(f + sg + . . .) where the perturbed eigenvector is normalised by 〈f + sg + . . . , f ∗ 〉 = 〈f, f ∗ 〉.4 A. ASLANYAN AND E. B. DAVIES We deduce tha... |

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Citation Context ...amily of non-self-adjoint operators parametrised by c ∈ C, c ̸= 0. Under suitable conditions the eigenvalues of these operators are known to be independent of c, and are called resonances of ˆ H; see =-=[7, 13]-=- for expositions of the theory of dilation analytic resonances. Since the operators ˆ Hc are unitarily equivalent for values of c with the same argument, we only consider c of the form c := e iθ/2 whe... |

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Numerical Methods
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Citation Context ... the described procedure is the simplest version of the boundary condition transfer, or pivotal condensation method we have chosen for the second order equation. There is an extensive literature (cf. =-=[1, 4, 11]-=- etc.) on the transfer methods where much more advanced techniques are developed. Although there are other possibilities, in this paper we get satisfactory results implementing the above version. Fina... |

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4 |
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Citation Context ...al coupling constant as well as for many nonself-adjoint anharmonic oscillators. There is a rapidly growing literature on pseudospectral theory, which was invented to explore just such possibilities, =-=[5, 6, 8, 9, 10, 15, 16, 17, 18, 19, 20]-=-. The authors thank the Engineering and Physical Sciences Research Council for support under grant No. GR/L75443 12 A. ASLANYAN AND E. B. DAVIES In this paper we return to the same type of operator, ... |

2 | London A 455 - Soc - 1998 |

1 |
Lectures on Operator Theory and its Applications
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Citation Context ...al coupling constant as well as for many nonself-adjoint anharmonic oscillators. There is a rapidly growing literature on pseudospectral theory, which was invented to explore just such possibilities, =-=[5, 6, 8, 9, 10, 15, 16, 17, 18, 19, 20]-=-. The authors thank the Engineering and Physical Sciences Research Council for support under grant No. GR/L75443 12 A. ASLANYAN AND E. B. DAVIES In this paper we return to the same type of operator, ... |

1 | Pseudospectra of differential operators (Preprint - Davies - 1997 |

1 |
Introduction to Spectral Theory (Springer--Verlag
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(Show Context)
Citation Context ...amily of non-self-adjoint operators parametrised by c ∈ C, c ̸= 0. Under suitable conditions the eigenvalues of these operators are known to be independent of c, and are called resonances of ˆ H; see =-=[7, 13]-=- for expositions of the theory of dilation analytic resonances. Since the operators ˆ Hc are unitarily equivalent for values of c with the same argument, we only consider c of the form c := e iθ/2 whe... |

1 |
D F Griffiths and G A Watson, Numerical Analysis 1991 (Longman Sci
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(Show Context)
Citation Context ...al coupling constant as well as for many nonself-adjoint anharmonic oscillators. There is a rapidly growing literature on pseudospectral theory, which was invented to explore just such possibilities, =-=[5, 6, 8, 9, 10, 15, 16, 17, 18, 19, 20]-=-. The authors thank the Engineering and Physical Sciences Research Council for support under grant No. GR/L75443 12 A. ASLANYAN AND E. B. DAVIES In this paper we return to the same type of operator, ... |

1 |
SIAM Review 39
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Citation Context |