共查询到20条相似文献,搜索用时 31 毫秒
1.
Patrick J. Browne 《Journal of Mathematical Analysis and Applications》1977,60(1):259-273
In this paper we continue our investigation of multiparameter spectral theory. Let H1,…, Hk be separable Hilbert spaces and H = ?r = 1kHr, be their tensor product. In each space Hr we have densely defined self-adjoint operators Tr and continuous Hermitian operators Vrs. The multiparameter eigenvalue problem concerns eigenvalues λ = (λ1,…, λn) ?Rk and eigenvectors such that . We develop a spectral theory for such systems leading to a Parseval equality and generalized eigenvector expansion. The results are applied to a k × k system of linked secondorder differential equations. 相似文献
2.
Matania Ben-Artzi 《Journal of Differential Equations》1980,38(1):51-60
Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = VS(r) + VL. Let λ = lim supr→∞VL(r) < ∞ (we allow λ = ? ∞) and set λ+ = max(λ, 0). Assume that for some r0, VL(r) ?C2k(r0, ∞) and that there exists δ > 0 such that . Assume further that and that 2kδ > 1. It is shown that: (a) The restriction of H to C∞(Rn) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous. 相似文献
3.
Patrick J Browne 《Journal of Differential Equations》1977,23(2):285-292
In this paper we study the linked nonlinear multiparameter system , where xr? [ar, br], yr is subject to Sturm-Liouville boundary conditions, and the continuous functions ars satisfy ¦ . Conditions on the polynomial operators Mr, Prs are produced which guarantee a sequence of eigenfunctions for this problem yn(x) = Πr=1kyrn(xr), n ? 1, which form a basis in . Here [a, b] = [a1, b1 × … × [ak, bk]. 相似文献
4.
Stanley J Benkoski 《Journal of Number Theory》1976,8(2):218-223
If r, k are positive integers, then denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r = 1. An explicit formula for is derived and it is shown that .If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2…psas; pi ∈ S and ai ≥ 0 for all i} and denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for are derived and it is shown that . 相似文献
5.
David L Russell 《Journal of Mathematical Analysis and Applications》1982,87(2):528-550
We suppose that K is a countable index set and that is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L2[a, b], a < b. Let Σ = {σ1, σ2,…, σm} consist of m complex numbers not in Λ. Then, with p(λ) = Πk = 1m (λ ? σk), forms a Riesz (strong) bas Sobolev space Hm[a, b]. If we take σ1, σ2,…, σm to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tneλkt. 相似文献
6.
Arthur Lubin 《Journal of Functional Analysis》1974,17(4):388-394
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral = ⊕L2(vt) dm(t) and the operator on , where e(s, t) = exp ∫st ∫Tdvλ(θ) dm(λ). Let μt be the measure defined by for all continuous ?, and let ?t(z) = exp[?∫ (eiθ + z)(eiθ ? z)?1dμt(gq)]. Call {vt} regular iff for all for 1 a.e. 相似文献
7.
Steven Zelditch 《Journal of Functional Analysis》1983,50(1):67-80
We prove a Szegö-type theorem for some Schrödinger operators of the form with V smooth, positive and growing like . Namely, let πλ be the orthogonal projection of L2 onto the space of the eigenfunctions of H with eigenvalue ?λ; let A be a 0th order self-adjoint pseudo-differential operator relative to Beals-Fefferman weights and with total symbol a(x, ξ); and let f∈C(). Then (assuming one limit exists). 相似文献
8.
Teruo Ikebe 《Journal of Functional Analysis》1975,20(2):158-177
A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R3, is obtained, where V(x) is a long-range potential: , grad , being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator from PL2(R3) onto being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ), . 相似文献
9.
J.E Nymann 《Journal of Number Theory》1975,7(4):406-412
Given a set S of positive integers let denote the number of k-tuples 〈m1, …, mk〉 for which and (m1, …, mk) = 1. Also let denote the probability that k integers, chosen at random from , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then if k ≥ 3 and where d(S) denotes the natural density of S. From this result it follows immediately that as n → ∞. This result generalizes an earlier result of the author's where and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then as n → ∞. 相似文献
10.
Stanisław Lewanowicz 《Journal of Computational and Applied Mathematics》1979,5(3):193-206
In this paper we are constructing a recurrence relation of the form for integrals (called modified moments) in which Ck(λ) is the k-th Gegenbauer polynomial of order , and f is a function satisfying the differential equation of order n, where p0, p1, …, pn ? 0 are polynomials, and mk〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense. 相似文献
11.
D de Caen 《Journal of Combinatorial Theory, Series B》1983,34(3):340-349
The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let Xs(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(Xl), given that min(Xl) ≥ 1. The following lower bound on the variance of Xs is proved: , where m = |E(H)| and . This implies the following: putting t(k, l) = limn→∞T(n, l, k)(kn)?1 then t(k, l) ≥ T(s, l, k)((ks) ? 1)?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned. 相似文献
12.
Let λ1 and λN be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(hij), and x=(x1,x2,…,xN) with (x,x)=1. Then, it is known that (1) λ1?(x,Hx)?λN and (2) if, in addition, H is positive definite, . Assuming that y=(y1,y2,…, yN) and |yi|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H?1 are, respectively, replaced by the Hadamard products and , where M(y) is a matrix defined by . Subsequently, these results are extended to improve the spectral bounds of . 相似文献
13.
Matania Ben-Artzi 《Journal of Differential Equations》1980,38(1):41-50
Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = VS(r) + VL(r). Assume that for some r0, VL(r) ?C2k(r0, ∞) and that there exist μ > 0, δ > 0, such that . Assume further that min(2kμ, (2k ? 1)δ + μ) > 1. Under this weak decay condition on VL(r) it is shown in this paper that the positive spectrum of H is absolutely continuous and that the absolutely continuous part of H is unitarily equivalent to ?Δ, provided that the singularity of V at 0 is properly restricted. In particular, some oscillation of VL(r) at infinity is allowed. 相似文献
14.
S.K. Bajpai Joseph Tanne Donald Whittier 《Journal of Mathematical Analysis and Applications》1974,48(3):736-742
Let f(z), an analytic function with radius of convergence R (0 < R < ∞) be represented by the gap series ∑k = 0∞ckzλk. Set and define the growth constants ?, λ, T, t by , and if 0 < ? < ∞, . Then, assuming 0 < t < T < ∞, we obtain a decomposition theorem for f(z). 相似文献
15.
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
16.
We prove global well-posedness results for small initial data in , and in , sk=1/2?1/k, for the generalized Benjamin–Ono equation . We also consider the cases k=2,3. To cite this article: L. Molinet, F. Ribaud, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
17.
The probability measure of X = (x0,…, xr), where x0,…, xr are independent isotropic random points in n (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of . Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω⊥ with ‘initial’ point at the origin [ω⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and , where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in n invariant under the group of rotations in n, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x0,…, xr} for 1 ≤ r ≤ n. 相似文献
18.
J.H Michael 《Journal of Mathematical Analysis and Applications》1981,79(1):203-217
We consider the mixed boundary value problem , where Ω is a bounded open subset of n whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on . The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset of the reals such that, if then for is a Fredholm operator if and only if s ∈ . Moreover, = ?xewx, where the sets x are determined algebraically by the coefficients of the operators at x. If n = 2, x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, x is either an open interval of length 1 or is empty; and finally, if n ? 4, x is an open interval of length 1. 相似文献
19.
Douglas N. Clark 《Journal of Functional Analysis》1973,14(3):269-280
The operator acting on H=∝02π⊕L2(vt), where m and vt, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝ts∝Tdvλ(θ) dm(λ)], satisfies . It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence. 相似文献
20.
Richard Lavine 《Journal of Functional Analysis》1973,12(1):30-54
Absolute continuity in (0, ∞) for Schrödinger operators ? Δ + V(x), with long range potential V = V1 + V2 such that , ? > 0, as , is shown by proving estimates on resolvents near the real axis. Completeness of the modified wave operators for a superposition of Coulomb potentials also follows. Singular local behavior of V is allowed. 相似文献