## SPECTRAL BOUNDS USING HIGHER ORDER NUMERICAL RANGES (2004)

### BibTeX

@MISC{Davies04spectralbounds,

author = {E B Davies},

title = {SPECTRAL BOUNDS USING HIGHER ORDER NUMERICAL RANGES},

year = {2004}

}

### OpenURL

### Abstract

We describe how to obtain bounds on the spectrum of a non-self-adjoint operator by means of what we call its higher order numerical ranges. We prove some of their basic properties and describe explain how to compute them. We finally use them to obtain new spectral insights into the non-selfadjoint Anderson model in one and two space dimensions.

### Citations

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(Show Context)
Citation Context ...g their relationship with the pseudospectra of the operator concerned. The theory of pseudospectra is by now a well-established part of spectral analysis; we summarize the basic definitions, but cite =-=[3, 4, 30, 31]-=- for proofs and further information. If A is a bounded operator on H we define its pseudospectra as the family of sets Spec ε(A) = Spec(A) ∪ {z : ‖(zI − A) −1 ‖ > ε −1 } (8) parametrized by ε > 0. One... |

114 |
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Citation Context ...ue of If (H + iI) −1 is also compact then S = {x + iy : y 2 ≤ −σ(x)} Kx = (xI − H) 2 − B ∗ B. S = ⋃ ‖C‖≤1 Spec(H + CB) Proof We use standard variational methods for self-adjoint operators freely; see =-=[9]-=-. In particular we define the self-adjoint operator Kx to be that associated with the closed, semibounded, quadratic form Qx(f) = ‖(xI − H)f‖ 2 − ‖Bf‖ 2 whose form domain is Dom(H). We do not need to ... |

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38 |
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Citation Context ...te value of c. Example 15 We consider a discretization of the Airy operator defined on L 2 (−1, 1). This operator arises as a special case of the Orr-Sommerfeld problem and in the Torrey equation; see=-=[26, 27, 28, 29]-=-. Let Hf(x) = −h 2 f ′′ (x) subject to Dirichlet boundary conditions and let B = iV where V is the operator of multiplication by the function V (s) = s for all s ∈ (−1, 1). Redparth [27] has shown tha... |

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Citation Context ... 2 − ‖V ‖ 2 ≥ t(1 − ‖V ‖ 2 /t 2 ). Since δn = ‖V ‖ 2 /t, this is equivalent to the statement of the lemma. 4 The NSA Anderson Model The NSA Anderson model was introduced by Hatano, Nelson and others, =-=[7, 19, 20, 24]-=-; applications to non-hermitian quantum mechanics and to the growth of bacteria in an inhomogeneous environment were described. However, its purely mathematical properties turned out to be so interest... |

14 |
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Citation Context ...d only consider polynomials with leading coefficient 1 and constant coefficient 0. The polynomial numerical ranges just defined are obviously contained in the polynomial numerical hulls of Nevanlinna =-=[13, 18, 25]-=-, which we define by and The fact that Hull(p, A) := {z : |p(z)| ≤ ‖p(A)‖} Hulln(A) = ⋂ deg(p)≤n Hull(p, A). Numn(A) = Hulln(A) (4) for all n and all bounded A is far from obvious and was proved by Bu... |

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Citation Context ...op further in this paper, [23]. One would expect to be able to investigate the infinite volume limit using pseudospectral theory, and this has been done for a simple exactly soluble bidiagonal model, =-=[5, 14, 32]-=-. Our goal here is to consider this issue from yet another point of view, and to demonstrate that the infinite volume spectrum may be relevant to quite small finite systems. Whether or not this is tru... |

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Citation Context ... one extreme are potentials which are constant or periodic with period 2 and at the other random potentials which take independent values at each point. Several of the figures are similar to those in =-=[7, 19, 20, 24]-=- but we compare the results with our rigorous bounds, and also describe some phenomena which are new even at the numerical level. If the values ±γ of the potential at neighbouring points are strongly ... |

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Citation Context ...te value of c. Example 15 We consider a discretization of the Airy operator defined on L 2 (−1, 1). This operator arises as a special case of the Orr-Sommerfeld problem and in the Torrey equation; see=-=[26, 27, 28, 29]-=-. Let Hf(x) = −h 2 f ′′ (x) subject to Dirichlet boundary conditions and let B = iV where V is the operator of multiplication by the function V (s) = s for all s ∈ (−1, 1). Redparth [27] has shown tha... |

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(Show Context)
Citation Context ... directly as a bounded linear operator acting on l 2 (Z D ) for any D, and found that the spectrum consisted of a bounded region in the complex plane, which was described in some detail when D = 1 in =-=[10, 11]-=-. Nevertheless, a number of spectral questions needed to be clarified. Martínez has made substantial progress in resolving these using the methods which we develop further in this paper, [23]. One wou... |

6 |
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Citation Context ...A) makes sense if A is unbounded, but Hulln(A) does not. We mention that although we call Num2(A) the quadratic numerical range of A, it has no obvious connection with the concept of the same name in =-=[21, 22]-=-. They define a set W 2 (A) ⊆ C, but only for a 2 × 2-block operator matrix A. Their set can only have one or two components, whereas Num2(A) may have many: for example if A = A ∗ then Num2(A) = Spec(... |

5 |
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Citation Context ...op further in this paper, [23]. One would expect to be able to investigate the infinite volume limit using pseudospectral theory, and this has been done for a simple exactly soluble bidiagonal model, =-=[5, 14, 32]-=-. Our goal here is to consider this issue from yet another point of view, and to demonstrate that the infinite volume spectrum may be relevant to quite small finite systems. Whether or not this is tru... |

4 |
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Citation Context ...op further in this paper, [23]. One would expect to be able to investigate the infinite volume limit using pseudospectral theory, and this has been done for a simple exactly soluble bidiagonal model, =-=[5, 14, 32]-=-. Our goal here is to consider this issue from yet another point of view, and to demonstrate that the infinite volume spectrum may be relevant to quite small finite systems. Whether or not this is tru... |

4 | B: Spectral theory of pseudo-ergodic operators
- Davies
- 2001
(Show Context)
Citation Context ... directly as a bounded linear operator acting on l 2 (Z D ) for any D, and found that the spectrum consisted of a bounded region in the complex plane, which was described in some detail when D = 1 in =-=[10, 11]-=-. Nevertheless, a number of spectral questions needed to be clarified. Martínez has made substantial progress in resolving these using the methods which we develop further in this paper, [23]. One wou... |

4 |
The limit behaviour of the spectrum for large parameter values in a model problem
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Citation Context ...te value of c. Example 15 We consider a discretization of the Airy operator defined on L 2 (−1, 1). This operator arises as a special case of the Orr-Sommerfeld problem and in the Torrey equation; see=-=[26, 27, 28, 29]-=-. Let Hf(x) = −h 2 f ′′ (x) subject to Dirichlet boundary conditions and let B = iV where V is the operator of multiplication by the function V (s) = s for all s ∈ (−1, 1). Redparth [27] has shown tha... |

3 |
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(Show Context)
Citation Context ... 2 − ‖V ‖ 2 ≥ t(1 − ‖V ‖ 2 /t 2 ). Since δn = ‖V ‖ 2 /t, this is equivalent to the statement of the lemma. 4 The NSA Anderson Model The NSA Anderson model was introduced by Hatano, Nelson and others, =-=[7, 19, 20, 24]-=-; applications to non-hermitian quantum mechanics and to the growth of bacteria in an inhomogeneous environment were described. However, its purely mathematical properties turned out to be so interest... |

3 |
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(Show Context)
Citation Context |

2 |
A A, Aslanyan A, Davies E B: Bounds on complex eigenvalues and resonances
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Citation Context ...ately relies upon estimating B(zI − H) −1 in the L 1 operator norm and leads, among many much more general results, to the formula Spec(−∆ + V ) ⊆ {z : |z| ≤ ‖V ‖ 2 1/4} 9in one space dimension. See =-=[1, 2, 12]-=- for various derivations of this bound. The other method involves estimates of the Hilbert-Schmidt norm of B(zI − H) −1 . Both can be extended to L 2 (R N ) in principle, but the results are not so us... |

2 | A: On polynomial numerical hulls of normal matrices - Ch, Salemi - 2004 |

1 |
M S P: Analytic continuation and resonance free regions for Sturm-Liouville potentials with power decay
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Citation Context ...ately relies upon estimating B(zI − H) −1 in the L 1 operator norm and leads, among many much more general results, to the formula Spec(−∆ + V ) ⊆ {z : |z| ≤ ‖V ‖ 2 1/4} 9in one space dimension. See =-=[1, 2, 12]-=- for various derivations of this bound. The other method involves estimates of the Hilbert-Schmidt norm of B(zI − H) −1 . Both can be extended to L 2 (R N ) in principle, but the results are not so us... |

1 |
A: Some equivalent characterizations of the polynomial numerical hull of degree k
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(Show Context)
Citation Context ... that Hull(p, A) := {z : |p(z)| ≤ ‖p(A)‖} Hulln(A) = ⋂ deg(p)≤n Hull(p, A). Numn(A) = Hulln(A) (4) for all n and all bounded A is far from obvious and was proved by Burke and Greenbaum very recently, =-=[6]-=-. Note that Numn(A) makes sense if A is unbounded, but Hulln(A) does not. We mention that although we call Num2(A) the quadratic numerical range of A, it has no obvious connection with the concept of ... |

1 |
D R, Shnerb N M: Population dynamics and nonhermitian localization. pp 124
- Dahmen, Nelson
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(Show Context)
Citation Context ... 2 − ‖V ‖ 2 ≥ t(1 − ‖V ‖ 2 /t 2 ). Since δn = ‖V ‖ 2 /t, this is equivalent to the statement of the lemma. 4 The NSA Anderson Model The NSA Anderson model was introduced by Hatano, Nelson and others, =-=[7, 19, 20, 24]-=-; applications to non-hermitian quantum mechanics and to the growth of bacteria in an inhomogeneous environment were described. However, its purely mathematical properties turned out to be so interest... |

1 |
Khoruzhenko B A: Distribution of eigenvalues in nonHermitian Anderson
- Ya
- 1998
(Show Context)
Citation Context ...the operator on a finite interval subject to periodic or quasi-periodic boundary conditions, and proved results about the spectrum of the operator in the limit as the length of the interval diverged, =-=[15, 16, 17]-=-. They confirmed the earlier numerical studies indicating that the spectrum converges almost surely to a certain union of curves. They also obtained new results deriving rigorously the asymptotic dens... |

1 |
Khoruzhenko B A: Eigenvalue Curves of Asymmetric Tridiagonal Random Matrices
- Ya
(Show Context)
Citation Context ...the operator on a finite interval subject to periodic or quasi-periodic boundary conditions, and proved results about the spectrum of the operator in the limit as the length of the interval diverged, =-=[15, 16, 17]-=-. They confirmed the earlier numerical studies indicating that the spectrum converges almost surely to a certain union of curves. They also obtained new results deriving rigorously the asymptotic dens... |

1 |
Khoruzhenko B A: Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model
- Ya
- 2003
(Show Context)
Citation Context ...the operator on a finite interval subject to periodic or quasi-periodic boundary conditions, and proved results about the spectrum of the operator in the limit as the length of the interval diverged, =-=[15, 16, 17]-=-. They confirmed the earlier numerical studies indicating that the spectrum converges almost surely to a certain union of curves. They also obtained new results deriving rigorously the asymptotic dens... |

1 |
Matsaev V, Tretter C: A new concept for block operator matrices: The quadratic numerical range
- Langer, Markus
(Show Context)
Citation Context ...A) makes sense if A is unbounded, but Hulln(A) does not. We mention that although we call Num2(A) the quadratic numerical range of A, it has no obvious connection with the concept of the same name in =-=[21, 22]-=-. They define a set W 2 (A) ⊆ C, but only for a 2 × 2-block operator matrix A. Their set can only have one or two components, whereas Num2(A) may have many: for example if A = A ∗ then Num2(A) = Spec(... |

1 |
The non-self-adjoint Anderson Model
- Martínez
- 2004
(Show Context)
Citation Context ...the author of the spectrum of the non-self-adjoint (NSA) Anderson model. The result is two papers. In this one we prove some general theorems about the quadratic and higher order numerical ranges. In =-=[23]-=- Martínez proves surprisingly detailed results for the NSA Anderson model in one space dimension using quadratic numerical ranges. For some values of the parameters she is able to determine the spectr... |

1 |
Embree M: Spectra and Pseudospectra. Monograph to be published by Princeton Univ
- Trefethen
- 2004
(Show Context)
Citation Context ...e independent groups, and illustrate different aspects, or applications, of the ideas. In Section 2 we describe the relationship between polynomial numerical ranges and pseudospectra, as described in =-=[31]-=-. Section 3 obtains uniform bounds on the spectra of a structured family of (large, non-self-adjoint) perturbations of a given self-adjoint operator; applications to Schrödinger operators with complex... |