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12
Nonselfadjoint differential operators
, 2002
"... We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely ..."
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Cited by 30 (6 self)
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We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.
Nonselfadjoint harmonic oscillator, compact semigroups and pseudospectra
 J. Operator Theory
"... We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. ..."
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Cited by 10 (2 self)
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We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of HilbertSchmidt type in a maximal angular region. In order to show this last property, we deduce a nonselfadjoint version of the classical Mehler’s formula.
Spectral properties of random nonselfadjoint matrices and operators
, 2001
"... We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also desc ..."
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Cited by 8 (4 self)
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We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the nonselfadjoint Anderson model changes suddenly as one passes to the infinite volume limit.
Spectral theory of pseudoergodic operators
 Commun. Math. Phys
, 2001
"... We define a class of pseudoergodic nonselfadjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a nonselfadjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 t ..."
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Cited by 4 (3 self)
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We define a class of pseudoergodic nonselfadjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a nonselfadjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators.
Spectral theory of ordinary and partial linear differential operators on finite intervals
"... ii A new, unified transform method for boundary value problems on linear and integrable nonlinear partial differential equations was recently introduced by Fokas. We consider initialboundary value problems for linear, constantcoefficient evolution equations of arbitrary order on a finite domain. We ..."
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Cited by 3 (2 self)
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ii A new, unified transform method for boundary value problems on linear and integrable nonlinear partial differential equations was recently introduced by Fokas. We consider initialboundary value problems for linear, constantcoefficient evolution equations of arbitrary order on a finite domain. We use Fokas ’ method to fully characterise wellposed problems. For odd order problems with nonRobin boundary conditions we identify sufficient conditions that may be checked using a simple combinatorial argument without the need for any analysis. We derive similar conditions for the existence of a series representation for the solution to a wellposed problem. We also discuss the spectral theory of the associated linear twopoint ordinary differential operator. We give new conditions for the eigenfunctions to form a complete system, characterised in terms of initialboundary value problems. Acknowledgements ACKNOWLEDGEMENTS iii My first thanks go to my supervisor, Beatrice Pelloni. This work depends upon the guidance and encouragement she offered. Her scholarship and dedication has been an inspiration in my studies.
EIGENVALUES OF AN ELLIPTIC SYSTEM
, 2001
"... We describe the spectrum of a nonselfadjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilber ..."
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We describe the spectrum of a nonselfadjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics. While our study makes no claim to generality, the results obtained will have to be incorporated into any future general theory.
Spectral theory of some nonselfadjoint linear differential operators.
, 2013
"... ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC. ..."
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Cited by 1 (0 self)
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ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC.
1 SPECTRAL ASYMPTOTICS OF THE NONSELFADJOINT HARMONIC OSCILLATOR
, 2003
"... We obtain an explicit asymptotic formula for the norms of the spectral projections of the nonselfadjoint harmonic oscillator H. We deduce that the spectral expansion of e −Ht is norm convergent if and only if t is greater than a certain explicit positive constant. 1 ..."
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We obtain an explicit asymptotic formula for the norms of the spectral projections of the nonselfadjoint harmonic oscillator H. We deduce that the spectral expansion of e −Ht is norm convergent if and only if t is greater than a certain explicit positive constant. 1
School of Mathematical and Physical Sciences
, 2013
"... Spectral theory of some nonselfadjoint linear differential operators by B. Pelloni and D.A. SmithSpectral theory of some nonselfadjoint linear differential operators ..."
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Spectral theory of some nonselfadjoint linear differential operators by B. Pelloni and D.A. SmithSpectral theory of some nonselfadjoint linear differential operators
1 ASYMPTOTIC BEHAVIOUR OF QUASIORTHOGONAL POLYNOMIALS
, 2003
"... We obtain explicit upper and lower bounds on the norms of the spectral projections of the nonselfadjoint harmonic oscillator. Some of our results apply to a variety of other families of orthogonal polynomials. 1 ..."
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We obtain explicit upper and lower bounds on the norms of the spectral projections of the nonselfadjoint harmonic oscillator. Some of our results apply to a variety of other families of orthogonal polynomials. 1