(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) 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 half-line Schrödinger operators and the inverse transmission coefficient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known 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 Böttchers formula for the 2-modified 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 Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

Quasiexact Solution of a Relativistic Finite-Difference Analogue of the Schrödinger Equation for a Rectangular Potential Well

We consider a well-posed formulation of the spectral problem for a relativistic analogue of the one-dimensional Schrödinger equation with differential operators replaced with operators of finite purely imaginary argument shifts exp(±id/dx). We find effective solution methods that permit determining the spectrum and investigating the properties of wave functions in a wide parameter range for this problem in the case of potentials of the type of a rectangular well. We show that the properties of solutions of these equations depend essentially on the relation between and the parameters of the potential and a situation in which the solution for 1 is nevertheless fundamentally different from its Schrödinger analogue is quite possible.     

A limiting absorption principle for Schrödinger operators with Von Neumann-Wigner type potentials

In this paper we prove the main step in establishing a limiting absorption principle for von Neumann-Wigner type Schrödinger Hamiltonians of the form –+csinb|x|/|x|+V(x), whereV(x) is a short range potential. The first fundamental step is to obtain a limiting absorption principal for the free operator –+csinb.|x|/|x|. The free operator is unitarily equivalent to a direct sum of ordinary differential operators. We obtain uniform estimates for the resolvents of these ordinary differential operators. by obtaining uniform estimates for the Weyl-Green kernels of these resolvents. In turn, these latter estimates require uniform estimates on the Wronskians of certain generalized eigen-solutions of these differential operators.This paper is dedicated to the memory of the late Professor Charles C. Conley.     

Dissipative Schrödinger Operators with Matrix Potentials

Maximal dissipative Schrödinger operators are studied in L ²((–,);E) (dimE=n<) that the extensions of a minimal symmetric operator with defect index (n,n) (in limit-circle case at – and limit point-case at ). We construct a selfadjoint dilation of a dissipative operator, carry out spectral analysis of a dilation, use the Lax–Phillips scattering theory, and find the scattering matrix of a dilation. We construct a functional model of the dissipative operator, determine its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator and investigate its analytic properties. Finally, we prove a theorem on completeness of the eigenvectors and associated vectors of a dissipative Schrödinger operators.     

The complex germ method in fock space. I. Wave packet type asymptotics

In this paper, we establish a new method of constructing approximate solutions to secondary-quantized equations, for instance, for many-particle Schrödinger and Liouville equations written in terms of the creation and annihilation operators, and also for equations of quantum field theory. The method is based on transformation of these equations to an infinite-dimensional Schrödinger equation, which is investigated by semiclassical methods. We use, and generalize to the infinite-dimensional case, the complex germ method, which yields wave packet type asymptotics in the Schrödinger representation. We find the corresponding asymptotics in the Fock space and show that the state vectors obtained are actually asymptotic solutions to secondary-quantized equations with an accuracy of O(^M/2), M N, with respect to the parameter of the semiclassical expansion.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 310–329, August, 1995.     

A General Families Index Theorem

We consider the heat operator of a Bismut superconnection for a family of generalized Dirac operators defined along the leaves of a foliation with Hausdorff graph. We assume that the strong Novikov–Shubin invariants of the Dirac operators are greater than three times the codimension of the foliation. We compute the t asymptotics associated to a rescaling of the metric by 1/t and show that the heat operator converges to the Chern character of the index bundle of the operator. Combined with previous results, this gives a general families index theorem for such operators.     

Three formulas for eigenfunctions of integrable Schrödinger operators

We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherlandsintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the Etingof–Kirillov tracesof intertwining operators, the second an integral representationof hypergeometric type, and the third is a formula of Bethe ansatz type.The last two formulas are degenerations of elliptic formulasobtained previously in connection with theKnizhnik–Zamolodchikov–Bernardequation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctionsare parametrized by a Hermite–Bethe variety, a generalizationof the spectral variety of the Lamé operator.We also give the q-deformed version of ourfirst formula. In the scalar slN case, this gives common eigenfunctionsof the commuting Macdonald–Rujsenaars difference operators.     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) 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 half-line Schrödinger operators and the inverse transmission coe.cient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known 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 2-modified 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 Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone

The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t³ (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).     

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

The Schrödinger operator with a weakly accelerating potential

The ideas of scattering theory are applied to the construction of a unitary operator realizing the similarity of the operator - id/d in L₂() with a one-dimensional Schrödinger operator on the semiaxis with potential v(x), admitting at infinity the estimate.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 10–12, 1985.     

On the Singular Spectrum for Adiabatic Quasi-Periodic Schrödinger Operators on the Real Line

In this paper, we study spectral properties of a family of quasi-periodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent, and show that the spectrum is purely singular.Résumé. Cet article est consacré à létude du spectre dune famille dopérateurs de Schrödinger quasi-périodiques sur laxe réel lorsque les courbes iso-énergétiques adiabatiques sont non bornées dans la direction des moments. Dans des intervalles dénergies où cette propriété est vérifiée, nous obtenons une formule asymptotique pour lexposant de Lyapounoff, et nous démontrons que le spectre est purement singulier.Communicated by Bernard Helffersubmitted 17/06/03, accepted 05/03/04     

Singular Schrödinger Operators as Limits Point Interaction Hamiltonians

In this paper we give results on the approximation of (generalized) Schrödinger operators of the form - + µ for some finite Radon measure µ on R^d. For d = 1 we shall show that weak convergence of measures µ_n to µ implies norm resolvent convergence of the operators - + µ_n to - + µ. In particular Schrödinger operators of the form - + µ for some finite Radon measure µ can be regularized or approximated by Hamiltonians describing point interactions. For d = 3 we shall show that a fairly large class of singular interactions can be regarded as limit of point interactions.     

Spectral properties of one-dimensional disperse crystals

The connection between the spectral characteristics of the one-dimensional Schrödinger operator with periodic potential, asa, and the spectral characteristic of the Schrödinger operatorl(y)=–y+q(x)y with a decreasing potentialq(x) is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 197–211, 1984.     

Asymptotic behavior of solutions of the Korteweg-de Vries equation for large times

For the KdV equation a complete asymptotic expansion of the dispersive tail for large times is described, and generalized wave operators are introduced. The asymptotics for large times of the spectral Schrödinger equation with a potential of the type of a solution of the KdV equation is studied. It is shown that the KdV equation is connected in a specific manner with the structure of the asymptotics of solutions of the spectral equation. As a corollary, known explicit formulas for the leading terms of the asymptotics of solutions of the KdV equation in terms of spectral data corresponding to the initial conditions are obtained. A plan for justifying the results listed is outlined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 32–50, 1982.     

One Version of the Bethe–Sommerfeld Conjecture

We consider the two-dimensional periodic Schrödinger operator under the assumption that the electric potential contains a term proportional to the -function concentrated on a periodic system of orthogonal lines. For this operator we confirm the Bethe–Sommerfeld conjecture and study the asymptotic behavior of the integrated density of states. We prove that the -potential can be chosen in such a way that the spectrum of the operator contains any given number of gaps. Bibliography: 9 titles.     

Integrated Density of States for Ergodic Random Schrödinger Operators on Manifolds

We consider the Riemannian universal covering of a compact manifold M = X/ and assume that is amenable. We show the existence of a (nonrandom) integrated density of states for an ergodic random family of Schrödinger operators on X.     

Theory of potential scattering,taking into account spatial anisotropy

We get new tests for the existence and completeness of wave operators under perturbation of a pseudodifferential operator with constant symbol P() by a bounded potential v(x). The term anisotropic is understood in the sense that the growth of P() as and the decrease of v(x) as x can depend essentially on the direction of the vectors and x respectively. This permits us to include in the sphere of applications of the abstract scattering theory of a nonelliptic unperturbed operator the D'Alembert operator, an ultrahyperbolic operator, nonstationary Schrödinger operator, etc. In view of the anisotropic character of the assumptions on the potential, the results obtained are new even in the elliptic case. As an example we consider a Schrödinger operator with potential close to the energy of a pair of interacting systems of many particles.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 35–51, 1977.     

On properties of the discrete and continuous spectrum for the radial dirac equation

To characterize the Dirac radial equation spectrum, we introduce the notion of a quantum defect _k, which is a generalization of this notion for the Schrödinger radial equation. The existence of _k is proved and explicit formulas for calculating _k are found for a broad class of potentials.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 36–49, July, 1996.