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1.
The scattering problem for the matrix Schrödinger operator with a non-Hermitian potential is considered. It is shown that there exists a set of unsymmetric potentials for which the Wronskian can be introduced. For a real k, an explicit expression for the Wronskian is derived. For a complex k, the asymptotic value of the Wronskian, as x± , is determined.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 133–139.  相似文献   

2.
Let be a lattice in the n-dimensional Euclidean space Rn and let F be the fundamental domain of the lattice . We denote by H the Schrödinger operator generated in L2(Rn) by the expression –u + q(x)u(1), and by Ht the operator generated in L2(F) by the expression (1) and by quasiperiodic boundary conditions, where q(x) is a periodic (with respect to the lattice ) function. Asymptotic formulas for the eigenvalues of the operator Ht are obtained and with the aid of these formulas it is proved that there exists a number (q) such that the interval [(q), ] belongs to the spectrum of the operator H [for n3 in the case of sufficiently smooth potentials q(x), while for n=2 for any potential q(x) from L2(F)], i.e., the Bethe-Sommerfeld conjecture is proved for arbitrary lattices.Translated from Teoriya Funktsii, Funktsionali'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 17–34, 1988.  相似文献   

3.
Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cn and let fL2(D). For the Hankel operator Hf on the Bergman space A2(D), it is shown that the essential norm of Hf in L2(D) is comparable to the distance norm from Hf to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).  相似文献   

4.
In this paper, the -subdifferential is introduced for investigating the global behavior of real-valued functions on a normed spaceX. Iff: DX attains its global minimum onD atx *, then 0 f(x *). This necessary condition always holds, even iff is not continuous orx * is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable +, many local minima cannot satisfy this necessary condition. For the sufficient conditions, the so-called -convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a -convex function. There are -convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two -convex functions. For all that, -convex functions still have properties similar to those of convex functions. For instance, each -local minimizer off is at the same time a global one. Iff attains its global minimum onD, then it does so at least at one point of its -boundary.This research was supported by the Alexander von Humboldt Foundation. The author thanks Professors R. Bulirsch, K. H. Hoffmann, and H. G. Bock for inviting him to Munich and Augsburg where this research was done.  相似文献   

5.
We study spectral and scattering properties of the Laplacian H (σ)=-Δ in corresponding to the boundary condition with a periodic function σ. For non-negative σ we prove that H (σ) is unitarily equivalent to the Neumann Laplacian H (0). In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of H (σ) under mild assumptions on σ.  相似文献   

6.
The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .  相似文献   

7.
One considers the total scattering cross section on the potential gV(x), xm, m3, for large values of the coupling constant g and of the wave number k. One assumes that V(x)(x/|1x|)|x|, 2>m+1, as ¦x¦. It is shown that for gk–1 , g3–ak2(a–2) the scattering cross section is equal asymptotically to a(gk–1), x=(m–1)(–1)–1. Here the coefficient a is determined only by the function and the number . Under the additional conditions >0, V>0, the indicated asymptotic behavior holds in the large domain gk–1 , gka–z c(gk–1), >0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 105–136, 1986.  相似文献   

8.
The Dirac equation, as a 4×4-hyperbolic system on 3, possesses an invariant algebra of global pseudodifferential operators-in the sense that conjugation with the Dirac time propagator leaves the algebra invariatn (cf. [CX]. Chapter 10). In this paper we examine the relation between the two invariant algebras att=0 and att'=0 when (t,x) and (t',x') are coordinates of Minkowsky space related by a (proper) Lorentz transform. For vanishing electromagnetic potentials these algebras are transforms of each other by the implied change of dependent and independent variables. In the general case such a space-time transform will make the potentials time dependent, hence also the algebra dependent on the initial plane.  相似文献   

9.
A formal method of constructing the viscosity solutions for abstract nonlinear equations of Hamilton–Jacobi–Bellman (HJB) type was developed in the previous work of the author. A new advantage of this method (which was called an nonlinear potentials method) is that it gives a possibility to choose at the first step an expected regularity of the solution and then – to construct this solution. This makes the whole procedure more simple because an analysis of regularity of viscosity solutions is usually the most complicated step.Nonlinear potentials method is a generalization of Krylov's approach to study HJB equations.In this article nonlinear potentials method is applied to elliptic degenerate HJB equations in Rd with variable coefficients.  相似文献   

10.
For a fixed value of the spectral parameter (energy) one derives the asymptotic behavior with respect to a large index of the eigenvalue of the scattering matrix. The role of the unperturbed operator is played by a pseudodifferential operator (PDO) with constant coefficients in L2(m). The perturbed operator is obtained by the addition of a differential expression whose coefficients behave asymptotically at infinity as homogeneous functions of order. One considers also perturbations of the type medium perturbation. On the initial PDO and on the perturbations one imposes certain restrictions. The problem reduces to the investigation of the asymptotics of the spectrum of some PDO of order on a surface of constant energy. Typical examples are: different variants of the Schrodinger equations, the Dirac system, the Maxwell system.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 3–29, 1981.  相似文献   

11.
Let G and {kj} be the domain and the array mentioned in the title (the boundary of the domain is assumed to be rectifiable). One describes a general scheme for the approximation of fonctionsf in the domain G, representable in the form f(z)=(2i)–1g()(–z)–1d, where g Lz (G), by a sequence of rational fractions. The characteristic feature of this scheme is the fact that the poles k of the fraction lie in the k-th row of the array {kj}. There is given a condition on {kj}, necessary and sufficient in order that each functionf, of the kind described above, should admit a uniform approximation inside G with the aid of the indicated scheme. In the case when this condition is not satisfied and \G.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 254–273, 1989.  相似文献   

12.
For the difference of nonordinary renewal processes, we find the distribution of the main boundary functionals. For the queuing system D |D |1, we determine the distribution of the number of calls in transient and stationary modes.  相似文献   

13.
Properties of the scattering operator and associated semigroup generated by the generalized Lax-Phillips scheme are studied. In particular, it is proved that the scattering operator admits decomposition into an orthogonal sum S=S1 S2, where S1 is a unitary scattering operator in the sense of Lax-Phillips, S2 is a completely nonunitary operator of the type studied by C. Foias. For the associated semigroup T(t) up to the unitary component one gets the formula T(t)=T1(t) T2(t), where T1(t) is the associated semigroup of class C00 and there exist subspaces such that a) The study of the structure of the original space and the properties of the operator S and of the semigroup T(t) let one decompose the scattering matrix into the orthogonal sum of pure and constant unitary parts.Translated from Dinamicheskie Sistemy, No. 4, pp. 125–130, 1985.  相似文献   

14.
We consider the integral We solve the problem of determination of necessary and sufficient conditions in order that (u) be independent of the values of u(x) inside a bounded domain . These conditions are written in the form of a set of differential equations for the functions f(x,u,¯p,Tij) on the set m{x; u+¯p+ Tij<}. For such functions (u) is represented in the form of a boundary integral.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 35–51, 1975.  相似文献   

15.
In this paper, we show the existence of the solutions of the screen problems of the Stokes equation in R n , n ≥ 3. For the purpose of it, we use the fractional layer potentials (for the Stokes equation) in a bounded Lipschitz domain The author is supported by the Korea Research Grant Foundation Grant (MOEHRD)KRF-2005-214-C00179.  相似文献   

16.
Given a metrizable compact convex setX of a locally convex Hausdorff space, a positive projectionT:C(X, )C(X, ) and a continuous function :X[0, 1], it is shown that under suitable assumptions there exists a positive contraction semigroup onC(X, ) that can be represented in terms of the Lototsky-Schnabl operators associated withT and . Several properties of this semigroup are investigated. In particular, its infinitesimal generator is determined in a core of its domain. WhenX p for somep1, then the generator is shown to be a degenerate elliptic second order differential operator.Dedicated to Professor George Maltese on the occasion of his 60th birthday  相似文献   

17.
We prove that the boundary of a bounded domain is a set of injectivity for the twisted spherical means on n for a certain class of functions on n . As a consequence we obtain results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform.  相似文献   

18.
Let R be a subring of the rationals with 1/2, 1/3R; let S R n denote the R-local n-sphere and define R n :=S R n for n odd, R n :=S R n for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces R n (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M*(E;R); here M* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by *(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra *(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.  相似文献   

19.
A cutting plane method for linear programming is described. This method is an extension of Atkinson and Vaidya's algorithm, and uses the central trajectory. The logarithmic barrier function is used explicitly, motivated partly by the successful implementation of such algorithms. This makes it possible to maintain primal and dual iterates, thus allowing termination at will, instead of having to solve to completion. This algorithm has the same complexity (O(nL 2) iterations) as Atkinson and Vaidya's algorithm, but improves upon it in that it is a long-step version, while theirs is a short-step one in some sense. For this reason, this algorithm is computationally much more promising as well. This algorithm can be of use in solving combinatorial optimization problems with large numbers of constraints, such as the Traveling Salesman Problem.  相似文献   

20.
Fradon  Myriam 《Potential Analysis》1997,6(4):369-414
On a domain D in d, for a smooth enough probability density and a diffusion matrix which can degenerate, we construct the law Q s of a (x)d -symmetric reflecting process in D with matrix . Therefore, we use the associated Dirichlet form and a sequence of approximating processes already used by Pardoux and R. Williams in [23]. Under mild conditions on the boundary ofD (finite Minkowski content), we prove that Q s is the law of a semi-martingale and provide its decomposition. Comparing with the decomposition in additive functionals, we conclude that the process is reflected in the conormal direction * n where n denotes Chen's normal (cf [10]), that is, the reflection direction of the Brownian motion in Kuramochi compactification.  相似文献   

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