共查询到20条相似文献,搜索用时 31 毫秒
1.
Jerome A Goldstein James T Sandefur 《Journal of Mathematical Analysis and Applications》1979,67(1):58-74
Let H be a self-adjoint operator on a complex Hilbert space . The solution of the abstract Schrödinger equation is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥2 is independent of t. When does the energy break up into different kinds of energy E = ∑j = 1NEj(t) which become asymptotically equipartitioned ? (That is, for all j and all data u(0).) The “classical” case is the abstract wave equation self-adjoint on 1. This becomes a Schrödinger equation in a Hilbert space (essentially is two copies of 1), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C1-algebra approach to quantum field theory and statistical mechanics. (2) Let A1,…, AN be commuting self-adjoint operators with N = 2 or 4. Then the equation admits equipartition of energy if and only if exp(it(Aj ? Ak)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k. 相似文献
2.
Rudolf Wegmann 《Journal of Mathematical Analysis and Applications》1976,56(1):113-132
For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by · number {λi: λi ? x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, , p) with the same distribution Fa. Suppose that all moments | a | k, k = 1, 2, … are finite, a=0 and | a | 2. Let with complex numbers θσ and finite products Pσ of factors A and (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples , , , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form of Schur's inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n. 相似文献
3.
Ivan Singer 《Journal of Mathematical Analysis and Applications》1980,76(2):339-368
We show that, if (F →uX) is a linear system, a convex target set and a convex functional, then, under suitable assumptions, the computation of inf ) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of , or of the infima on F of Lagrangians, involving elements of . Also, we prove similar results for a convex system (F →uX) and the convex cone Ω of all non-positive elements in X. 相似文献
4.
In this paper we estimate the difference between the spectrum of A (which is supposed to be known completely) and A+E, in terms of the norms of E and . Also we find a special upper triangular form K = Q-1AQ which separates the distinct eigenvalues of A, and we estimate the 6Q-1626Q62 from the above. 相似文献
5.
We characterize the uniform algebras A on a compact Hausdorff space X which contain a sequence {uj}j = 0∞ of unimodular elements with and closed span in terms of the maximal ideal space of A. Roughly, the essential set of A looks like (at most) countably many copies of the boundary of the unit disk, and A looks like the disk algebra on each. 相似文献
6.
Let k be , or , and set . We compute K2(A) and K3(A). Our method is to construct a map and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K2(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?kA. 相似文献
7.
8.
Michel Talagrand 《Comptes Rendus Mathematique》2003,337(7):477-480
Consider a random Hamiltonian for We assume that the family is jointly Gaussian centered and that for =ξ(N?1∑i?Nσ1iσ2i) for a certain function ξ on . F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian is bounded below by the free energy of the Parisi solution provided that ξ is convex on . We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on . This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
9.
Let A be an n×n real matrix, and be a closed convex cone. The spectrum of A relative to K, denoted by σ(A,K), is the set of all for which the linear complementarity problem admits a nonzero solution . The aim of this Note is to study the main properties of the set-valued mapping , and discuss some structural differences existing between the polyhedral case (i.e., K is finitely generated) and the non-polyhedral case. To cite this article: A. Seeger, M. Torki, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
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11.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
12.
Z.A Karian 《Journal of Number Theory》1976,8(2):233-244
Let k be a positive square free integer, the ring of algebraic integers in and S the unit sphere in Cn, complex n-space. If A1,…, An are n linearly independent points of Cn then L = {u1Au + … + unAn} with is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3. 相似文献
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14.
It is shown that if satisfies , where σk(A) denotes the sum of all kth order subpermanent of A, then Per[λJn+(1?λ)A] is strictly decreasing in the interval 0<λ<1. 相似文献
15.
A Connes 《Advances in Mathematics》1981,39(1):31-55
In this paper we show the existence and uniqueness of a natural isomorphism øjα of Kj(A) with Kj+1(A ?α), j ? /2 where (A, , α) is a dynamical -system, K is the functor of topological K theory and A ?α is the crossed product of A by the action of . The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace gt, one has for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S3) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple algebra with no nontrivial idempotent. 相似文献
16.
In this Note we study the Schrödinger equation i?tu+Δu+V0u+V1u=0 on with initial condition , where V0 is a coulombian potential, singular at finite distance and V1 is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
17.
A regularity result for singular nonlinear elliptic systems in inverse-power weighted Sobolev spaces
P.D Smith 《Journal of Differential Equations》1984,53(2):125-138
The compactness method to weighted spaces is extended to prove the following theorem:Let H2,s1(B1) be the weighted Sobolev space on the unit ball in Rn with norm Let n ? 2 ? s < n. Let u? [H2,s1(B1) ∩ L∞(B1)]N be a solution of the nonlinear elliptic system , are uniformly continuous functions of their arguments and satisfy: . Then there exists an R1, 0 < R1 < 1, and an α, 0 < α < 1, along with a set such that (1) , (2) Ω does not contain the origin; Ω does not contain BR1, (3) is open, (4) u is ; u is LipαBR1. 相似文献
18.
Let A be a C1-algebra and X a Banach A-module. The module action of A on X gives rise to module actions of on and , and derivations of A into X (resp. ) extend to derivations of into (resp. ). If A is nuclear, and X is a dual Banach A-module with weakly sequentially complete, then every derivation of A into X is inner. Under the same hypothesis on A, the extension to the finite part of of any derivation of A into any dual Banach A-module is inner, as are all derivations of A into . Every derivation of a semifinite von Neumann algebra into its predual is inner. 相似文献
19.
In this paper, we establish the following results: Let A be a square matrix of rank r. Then (a) is idempotent of rank r, and trrA (defined as the sum of the principal minors of order r in A) is one iff A is Hermitian idempotent. (b) As=At for some positive integers s≠t, and trA=rankA iff A is idempotent. (c) for some integers s≠t iff is idempotent, while for some integers s≠0 iff . (d) for some integers s≠t and rankA=trA iff A is Hermitian idempotent, while for some integer s iff A is Hermitian. Here indicates the conjugate transpose of A, and P-α is defined iff (P+)α=(Pα)+ for all positive integers α and P+ is the Moore-Penrose inverse of P. 相似文献
20.
《Nonlinear Analysis: Theory, Methods & Applications》2004,56(2):185-199
In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation ut=div(um−1|∇u|p−2∇u)+Vum+p−2 in a cylinder , with initial condition u(·,0)=u0(·)⩾0 and vanishing on the parabolic boundary . Here (resp. ) is a bounded domain with smooth boundary, , , 1<p<N and m+p−2>0. The critical exponents are found and the nonexistence results are proved for . 相似文献