## (2008)

### Abstract

Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

### Citations

98 |
Spectral theory and differential operators
- Davies
- 1995
(Show Context)
Citation Context ...∗ c . Then, by Proposition 3.1, µ is an eigenvalue of L and hence real, since L is symmetric. Hence the deficiency indices of Lc are both zero, so Lc is essentially self adjoint (see Theorem 1.2.7 in =-=[6]-=-). Since L is a symmetric extension of Lc the result follows. 4 Compactness of the resolvent We define γ : [0, 1] → R ∪ {∞} by ∫ x γ(x) = p(t) −1 dt (13) 0 for all x ∈ [0, 1] and G : [0, 1] × [0, 1] →... |

13 |
A new type of instability: explosive disturbances in a liquid film inside a rotating horizontal cylinder
- Benilov, O’Brien, et al.
(Show Context)
Citation Context ..., Levitin and Marletta have proven in a recent paper [2] that a wider class of operators possess only real eigenvalues. The operator H was first studied by Benilov, O’Brien and Sazonov, who showed in =-=[3]-=- that the equation ∂f ∂t (1) = Hf (2) 1approximates the evolution of a liquid film inside a rotating horizontal cylinder. Davies showed in [4] that −iH has compact resolvent by considering the unitar... |

10 | An indefinite convection-diffusion operator with real spectrum
- Weir
(Show Context)
Citation Context ...s which accumulate only at infinity. MSC classes: 34Lxx; 76Rxx; 34B24 Keywords: spectrum, non-self-adjoint, self-adjoint, fluid mechanics, eigenvalue, Sturm-Liouville 1 Introduction In a recent paper =-=[1]-=-, we showed that the spectrum of the highly non-selfadjoint operator −iH is real, where H is the closure of the operator H0 on L2 (−π, π) defined by (H0f)(θ) = ε ∂ ∂θ ( sin(θ) ∂f ∂θ ) + ∂f ∂θ for any ... |

10 | An indefinite convection-diffusion operator
- Davies
- 2007
(Show Context)
Citation Context ...t studied by Benilov, O’Brien and Sazonov, who showed in [3] that the equation ∂f ∂t (1) = Hf (2) 1approximates the evolution of a liquid film inside a rotating horizontal cylinder. Davies showed in =-=[4]-=- that −iH has compact resolvent by considering the unitarily equivalent operator A on l2 (Z) defined by (Av)n = ε 2 n(n − 1)vn−1 − ε 2 n(n + 1)vn+1 + nvn (3) for all v ∈ Dom (A) := {v ∈ l 2 (Z) : Av ∈... |

9 | and D Pelinovsky: Spectrum of an non-self-adjoint operator associated with the cylindric heat equation
- Chugunova
- 2007
(Show Context)
Citation Context ... that A = A− ⊕ 0 ⊕ A+, (4) where A− and A+ are the restrictions of A to Z− and Z+ respectively, and that A− is unitarily equivalent to −A+. Eigenvalues of H or −iH have been calculated numerically in =-=[3, 4, 5]-=-, but until now it has not been proven rigorously that any non-zero eigenvalues exist. In this paper we prove rigorously that −iH has infinitely many eigenvalues which accumulate at ±∞. Our approach i... |

8 | A PTsymmetric periodic problem with boundary and interior singularities. arXiv:0801.0172 - Boulton, Levitin, et al. - 2008 |