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49
Nonselfadjoint operators, infinite determinants, and some applications
, 2005
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Jost functions and Jost solutions for Jacobi matrices, II. Decay and Analyticity
"... Abstract. We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Jost solution. We also prove L2 convergence of Szegő as ..."
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Cited by 33 (15 self)
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Abstract. We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Jost solution. We also prove L2 convergence of Szegő asymptotics on the spectrum. 1.
(Modified) Fredholm Determinants for Operators with MatrixValued SemiSeparable Integral Kernels Revisited
, 2008
"... We revisit the computation of (2modified) Fredholm determinants for operators with matrixvalued semiseparable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our ap ..."
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Cited by 28 (10 self)
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We revisit the computation of (2modified) Fredholm determinants for operators with matrixvalued semiseparable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants. We illustrate our approach by identifying classical objects such as the Jost function for halfline Schrödinger operators and the inverse transmission coefficient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the wellknown expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant. Finally, we rederive the explicit formula for the 2modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener–Hopf analog of Day’s formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.
Inverse Resonance Scattering on the Real Line
 Inverse Problems
, 2002
"... We consider the Schrodinger operator f + qf in L (R), with a real compactly supported potential q. We give the solution of two inverse problems (including characterization) : q ! f zeros of the reection coecient g and q ! f bound states and resonances g. We describe the set of "isores ..."
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Cited by 22 (10 self)
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We consider the Schrodinger operator f + qf in L (R), with a real compactly supported potential q. We give the solution of two inverse problems (including characterization) : q ! f zeros of the reection coecient g and q ! f bound states and resonances g. We describe the set of "isoresonance potentials", i.e., we obtain all potentials with the same resonances and bound states.
Derivatives of (modified) Fredholm determinants and stability of standing and travelling waves
 J. Math. Pures Appl
, 2008
"... Abstract. Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman–Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency z ..."
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Cited by 19 (7 self)
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Abstract. Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman–Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analyticallyvarying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semiseparable integral kernels, which include in particular the general onedimensional case, and for sums thereof, which latter possibility appears to offer applications in the multidimensional case. A second main result is to show that the multidimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [23] on successive Galerkin subspaces, giving a natural extension of the onedimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration. 1.
Schrödinger operators with complexvalued potentials and no resonances
 Duke Math Jour
"... Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophas ..."
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Cited by 15 (9 self)
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Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on R d. In odd dimensions d ≥ 3 we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is superexponentially decaying in time. 1.
Several complex variables and the distribution of resonances for potential scattering
 Commun. Math. Phys
"... Abstract. We study resonances associated to Schrödinger operators with compactly supported potentials on R d, d ≥ 3, odd. We consider compactly supported potentials depending holomorphically on a parameter z ∈ C m. For certain such families, for all z except those in a pluripolar set, the associated ..."
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Cited by 15 (8 self)
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Abstract. We study resonances associated to Schrödinger operators with compactly supported potentials on R d, d ≥ 3, odd. We consider compactly supported potentials depending holomorphically on a parameter z ∈ C m. For certain such families, for all z except those in a pluripolar set, the associated resonancecounting function has order of growth d. 1.
The resonance counting function for Schrödinger operators with generic potentials
 Math. Research Letters
"... Abstract. We show that the resonance counting function for a Schrödinger operator has maximal order of growth for generic sets of realvalued, or complexvalued, L ∞compactly supported potentials. 1. ..."
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Cited by 13 (7 self)
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Abstract. We show that the resonance counting function for a Schrödinger operator has maximal order of growth for generic sets of realvalued, or complexvalued, L ∞compactly supported potentials. 1.
Eigenvalues and resonances using the Evans function
 Discrete and Continuous Dynamical Systems 10 (2004
"... Abstract. In this expository paper, we discuss the use of the Evans function in finding resonances, which are poles of the analytic continuation of the resolvent. We illustrate the utility of the general theory developed in [13, 14] by applying it to two physically interesting test cases: the linea ..."
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Cited by 12 (3 self)
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Abstract. In this expository paper, we discuss the use of the Evans function in finding resonances, which are poles of the analytic continuation of the resolvent. We illustrate the utility of the general theory developed in [13, 14] by applying it to two physically interesting test cases: the linear Schrödinger operator and the linearization associated with the integrable nonlinear Schrödinger equation. 1. Introduction. Consider
Trace formulas for Schrödinger operators in connection with scattering theory for finitegap backgrounds
 in Spectral Theory and Analysis
, 2011
"... Abstract. We investigate trace formulas for onedimensional Schrödinger operators which are trace class perturbations of quasiperiodic finitegap operators using Krein’s spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg–de Vries hierarchy ..."
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Cited by 7 (5 self)
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Abstract. We investigate trace formulas for onedimensional Schrödinger operators which are trace class perturbations of quasiperiodic finitegap operators using Krein’s spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg–de Vries hierarchy in this class and relate them to the reflection coefficients via Abelian integrals on the underlying hyperelliptic Riemann surface. 1.