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18
Perturbations of OneDimensional Schrödinger Operators Preserving the Absolutely Continuous Spectrum
, 2001
"... We study the stability of the absolutely continuous spectrum of onedimensional Schrodinger operators [Hu](x) = \Gammau 00 (x) + q(x)u(x) with periodic potentials q(x). Specifically, it is proved that any perturbation of the potential, V 2 L 2 , preserves the essential support (and multiplicity ..."
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Cited by 29 (3 self)
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We study the stability of the absolutely continuous spectrum of onedimensional Schrodinger operators [Hu](x) = \Gammau 00 (x) + q(x)u(x) with periodic potentials q(x). Specifically, it is proved that any perturbation of the potential, V 2 L 2 , preserves the essential support (and multiplicity) of the absolutely continuous spectrum. This is optimal in terms of L p spaces and, for q j 0, it answers in the affirmative a conjecture of Kiselev, Last and Simon. By adding constraints on the Fourier transform of V , it is possible to relax the decay assumptions on V . It is proved that if V 2 L 3 and V is uniformly locally square integrable, then preservation of the a.c. spectrum still holds. If we assume that q j 0, still stronger results follow: if V 2 L 3 and V (k) is square integrable on an interval [k 0 ; k 1 ], then the interval [k 2 0 =4; k 2 1 =4] is contained in the essential support of the absolutely continuous spectrum of the perturbed operator. vi Contents Ackn...
Resonances in One Dimension and Fredholm Determinants
, 2000
"... We discuss resonances for Schrödinger operators in whole and halfline problems. One of our goals is to connect the Fredholm determinant approach of Froese to the Fourier transform approach of Zworski. Another is to prove a result on the number of antibound states namely, in a halfline problem the ..."
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Cited by 24 (1 self)
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We discuss resonances for Schrödinger operators in whole and halfline problems. One of our goals is to connect the Fredholm determinant approach of Froese to the Fourier transform approach of Zworski. Another is to prove a result on the number of antibound states namely, in a halfline problem there are an odd number of antibound states between any two bound states.
CMV matrices: Five years after
, 2007
"... CMV matrices are the unitary analog of Jacobi matrices; we review their general theory. ..."
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Cited by 22 (3 self)
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CMV matrices are the unitary analog of Jacobi matrices; we review their general theory.
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 20 (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, Integral Equations and Operator Theory 47
, 2003
"... Dedicated with great pleasure to Eduard R. Tsekanovskii on the occasion of his 65th birthday. Abstract. 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 ..."
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Cited by 13 (6 self)
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Dedicated with great pleasure to Eduard R. Tsekanovskii on the occasion of his 65th birthday. Abstract. 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. 1.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Sum rules and spectral measures of Schrödinger operatros with L 2 potentials
"... Abstract. Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a halfline Schrödinger operator with square integrable potential. 1. ..."
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Cited by 9 (2 self)
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Abstract. Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a halfline Schrödinger operator with square integrable potential. 1.
Bound States for Schrödinger Hamiltonians: Phase Space Methods and Applications
 Rev. Math. Phys
, 1996
"... Properties of bound states for Schrodinger operators are reviewed. These include: bounds on the number of bound states and on the moments of the energy levels, existence and nonexistence of bound states, phase space bounds and semiclassical results, the special case of central potentials, and appli ..."
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Cited by 8 (0 self)
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Properties of bound states for Schrodinger operators are reviewed. These include: bounds on the number of bound states and on the moments of the energy levels, existence and nonexistence of bound states, phase space bounds and semiclassical results, the special case of central potentials, and applications of these bounds in quantum mechanics of many particle systems and dynamical systems. For the phase space bounds relevant to these applications we improve the explicit constants. Dedicated to Andr'e Martin, an old friend, who "retired" recently in order to accomplish even more! 1 Introduction Consider the Schrodinger operator H = \Gamma\Delta + V (x) (1) on L 2 (R d ). It describes either the motion of a quantum particle of mass 1=2 in an external potential V or the relative motion of two particles having reduced mass 1=2 interacting via the potential V . Units have been chosen such that Planck's constant ¯ h equals to one. In the present review we want to report about estimat...
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 5 (3 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.