## E B: On eigenfunction approximations for typical non-selfadjoint Schrödinger operators (1999)

Citations: | 4 - 0 self |

### BibTeX

@TECHREPORT{Aslanyan99eb:,

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

title = {E B: On eigenfunction approximations for typical non-selfadjoint Schrödinger operators},

institution = {},

year = {1999}

}

### OpenURL

### Abstract

Abstract. We construct efficient approximations for the eigenfunctions of non-selfadjoint Schrödinger operators in one dimension. The same ideas also apply to the study of resonances of self-adjoint Schrödinger operators which have dilation analytic potentials. In spite of the fact that such eigenfunctions can have surprisingly complicated structures with multiple local maxima, we show that a suitable adaptation of the JWKB method is able to provide accurate global approximations to them.

### Citations

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(Show Context)
Citation Context ...precise, this notion includes a vast variety of methods, each associated with certain asymptotic formulae. The general asymptotic theory has been developed in a number of books among which monographs =-=[5, 7, 12, 14]-=- can be mentioned. There exists an extensive literature on numerical analysis of ODEs where purely numerical procedures are combined with and strengthened by appropriate JWKB methods. Examples of diff... |

322 |
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Citation Context ...ding operators H1 and ˜ H1 are self-adjoint and known to have complex resonances8 A. ASLANYAN AND E. B. DAVIES which are the eigenvalues of the original non-self-adjoint operators (see, for example, =-=[2]-=- where the theory of dilation analytic resonances is exposed). The spectra of Ho and Hc for sufficiently large b are proved to be very close to each other; the same can be said about the eigenfunction... |

151 |
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(Show Context)
Citation Context ...precise, this notion includes a vast variety of methods, each associated with certain asymptotic formulae. The general asymptotic theory has been developed in a number of books among which monographs =-=[5, 7, 12, 14]-=- can be mentioned. There exists an extensive literature on numerical analysis of ODEs where purely numerical procedures are combined with and strengthened by appropriate JWKB methods. Examples of diff... |

32 | Semi-classical states for non-self-adjoint Schrdinger operators
- Davies
(Show Context)
Citation Context ... [8, 9, 11]. Apart from this, one can take advantage of similar techniques for strictly analytical purposes, including spectral analysis of Schrodinger operators. Recent papers on the subject include =-=[3, 4, 6, 10]-=-. In the papers [3, 4] relevant JWKB-type formulae have been used to construct the semi-classical modes for non-self-adjoint Schrodinger operators in order to study their spectral properties. A major ... |

24 | the harmonic oscillator and complex resonances
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- 1999
(Show Context)
Citation Context ... [8, 9, 11]. Apart from this, one can take advantage of similar techniques for strictly analytical purposes, including spectral analysis of Schrodinger operators. Recent papers on the subject include =-=[3, 4, 6, 10]-=-. In the papers [3, 4] relevant JWKB-type formulae have been used to construct the semi-classical modes for non-self-adjoint Schrodinger operators in order to study their spectral properties. A major ... |

16 |
Asymptotic analysis
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(Show Context)
Citation Context ...precise, this notion includes a vast variety of methods, each associated with certain asymptotic formulae. The general asymptotic theory has been developed in a number of books among which monographs =-=[5, 7, 13]-=- can be mentioned. There exists an extensive literature on numerical analysis of ODEs where purely numerical procedures are combined with and strengthened by proper JWKB methods. Examples of different... |

14 | Spectral instability for some Schrödinger operators
- Aslanyan, Davies
(Show Context)
Citation Context ...h multiple local maxima, we show that a suitable adaptation of the JWKB method is able to provide accurate global approximations to them. 1. Introduction The origin of this study was our recent paper =-=[1]-=-, where we discovered that extreme spectral instability was typical for many non-self-adjoint Schrodinger operators, and also for complex resonances of self-adjoint Schrodinger operators with dilation... |

7 |
A complex WKB method for adiabatic problems
- Fedotov, Klopp
- 2001
(Show Context)
Citation Context ... [8, 9, 11]. Apart from this, one can take advantage of similar techniques for strictly analytical purposes, including spectral analysis of Schrodinger operators. Recent papers on the subject include =-=[3, 4, 6, 10]-=-. In the papers [3, 4] relevant JWKB-type formulae have been used to construct the semi-classical modes for non-self-adjoint Schrodinger operators in order to study their spectral properties. A major ... |

6 | Semiclassical asymptotics of the residues of the scattering matrix for shape resonances
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(Show Context)
Citation Context |

4 |
An Introduction to Phase-Integral Methods, (Metheun and Co
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- 1962
(Show Context)
Citation Context ...precise, this notion includes a vast variety of methods, each associated with certain asymptotic formulae. The general asymptotic theory has been developed in a number of books among which monographs =-=[5, 7, 12, 14]-=- can be mentioned. There exists an extensive literature on numerical analysis of ODEs where purely numerical procedures are combined with and strengthened by appropriate JWKB methods. Examples of diff... |

3 |
A high-precision study of anharmonic-oscillator spectra
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- 1999
(Show Context)
Citation Context ...ical analysis of ODEs where purely numerical procedures are combined with and strengthened by appropriate JWKB methods. Examples of different asymptotic formulae can, for instance, be found in papers =-=[8, 9, 11]-=-. Apart from this, one can take advantage of similar techniques for strictly analytical purposes, including spectral analysis of Schrodinger operators. Recent papers on the subject include [3, 4, 6, 1... |

3 |
Brändas E: Weyl’s theory and the method of complex rotation. A synthesis for a description of the continuous spectrum
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(Show Context)
Citation Context ...s. Namely, their lower eigenvalues are real (up to a chosen accuracy); starting from a certain number they turn sharply into the lower half-plane. This behaviour of the complex resonances observed in =-=[1, 12]-=- seems to be typical for a range of operators, and the results quoted here provide yet another numerical evidence of the fact. Below we quote the even eigenvalues of ˜ Hc calculated by means of the sa... |

1 |
Numerical investigations of the free axisymmetric electric oscillations of an ideally conducting oblate spheroid
- Konyukhova
- 1995
(Show Context)
Citation Context ...ical analysis of ODEs where purely numerical procedures are combined with and strengthened by appropriate JWKB methods. Examples of different asymptotic formulae can, for instance, be found in papers =-=[8, 9, 11]-=-. Apart from this, one can take advantage of similar techniques for strictly analytical purposes, including spectral analysis of Schrodinger operators. Recent papers on the subject include [3, 4, 6, 1... |