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1.
The n-body problem is formulated as a problem of functional analysis on a Hilbert space G whose elements are analytic functions of complex dynamical variables. It is assumed that the two-body interaction is local and spherically symmetric, and belongs to the two-particle space G. The n-body resolvent R(λ) is constructed with the help of Fredholm methods. The operator R(λ) on G is associated with a family of operators R(λ, ?) on L2 which are resolvents of closed linear operators H(?), the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) contains a set of parallel half-lines starting at the thresholds of scattering channels and making an angle 2? with the positive real axis. The half-lines are branch cuts of R(λ, ?), but matrix elements of R(λ, ?) can be continued analytically across these. The operator R(λ, ?) may have isolated poles. The location of these does not depend on ?. Each pole is associated with one or more eigenvectors of H(?) belonging to spaces G. There may be poles off the real axis, the location of a pole determining for which values of ? it is on the physical sheet of H(?). It is shown how poles off the real axis give rise to resonances in the scattering cross section, the shape of a resonance being as one would expect on the basis of a model in which the scattering takes place via a decaying compound state having an eigenvector of H(?) with complex energy as its wave function.  相似文献   

2.
If the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λp, φ) starting at the thresholds λp of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable Lp-spaces, each half-line Y(λp, φ) is associated with an operator P(λp, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λp, φ) does not pass through any two- or three-particle eigenvalues λλp when φ runs through some interval 0 < α ? φ ? β < π2. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λp, φ) that are sufficient for P(λp, φ)[H(φ) ? λp] e?2 to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λp + ze2, the resolvent R(λp + ze2,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λp, φ) is included.  相似文献   

3.
Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number M such that there exist M different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(pm, ? 2) = (pm ? 2)!, R(pm + 1, ? 3) = (pm ? 2)!, R(k, ? k ? 3) = k!2, R(k, ? 0) = k, R(pm, ? 1) = pm(pm ? 1), R(pm + 1, ? 2) = (pm + 1)pm(pm ? 1). The exact value of R(k, ? λ) is determined whenever k ? k0(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k0(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for λ = (k2) + O(klog k).  相似文献   

4.
We consider the self-adjoint analytic family of operators H(z) in L2(Rm) defined for z ? Sα = {z ∥ Arg z ¦ < α}, associated with the operator H = H(1) = H0 + V, where H0 = ?Δ and V is a dilation-analytic short-range potential. The analytic connection between the local wave and scattering operators associated with the operators H(ei?) is established. The scattering matrix S(?) of H has a meromorphic continuation S(z) to Sα with poles precisely at the resolvent resonances of H, and the local scattering operators of e?2i?H(ei?) have representations in terms of the analytically continued scattering matrix S(?ei?).  相似文献   

5.
Starting from a defining differential equation (??t) W(λ, t, u) = (λ(u ? t)p(t)) W(λ, t, u) of the kernel of an exponential operator Sλ(?, t) = ∫?∞ W(λ, t, u)?(u) du with normalization ∫?∞W(λ, t, u) du = 1, we determine Sλ for various p(t) including; for example, p(t) a quadratic polynomial, all the known exponential operators are recovered and some new ones are constructed. It is shown that all the exponential operators are approximation operators. Further approximation properties of these operators are discussed. For example, functions satisfying ∥ Sλ(?, t) ? ?(t)∥ = O(λ) are characterized. Several results of C. P. May are also improved.  相似文献   

6.
The authors consider irreducible representations π ? N? of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels Kφ(x, y) of the trace class operations πφ = ∝Nφ(n)πndn, regarding the π as modeled in L2(Rk) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels Kφ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rn, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rn → Rn with the following properties. The Fourier transforms F1φ = Kφ(x, y, λ) all factor through A to give “rationalized” Fourier transforms (u) such that ° A = F1φ. On the rationalized parameter space a function f(u) is of the form Fφ = f ? f is Schwartz class on Rn. If polynomial operators T?P(N) are transferred to operators T? on Rn such that F(Tφ) = T?(Fφ), P(N) is transformed isomorphically to P(Rn).  相似文献   

7.
Define coefficients (κλ) by Cλ(Ip + Z)/Cλ(Ip) = Σk=0l Σ?∈Pk (?λ) Cκ(Z)/Cκ(Ip), where the Cλ's are zonal polynomials in p by p matrices. It is shown that C?(Z) etr(Z)/k! = Σl=k Σλ∈Pl (?λ) Cλ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (?λ), ? ∈ Pk, k ≤ 3. Several identities involving the (?λ)'s are derived. These are used to derive explicit expressions for coefficients of Cλ(Z)l! in expansions of P(Z), etr(Z)k! for all monomials P(Z) in sj = tr Zj of degree k ≤ 5.  相似文献   

8.
For a symmetric space GK of compact type, the highest-weight vectors for representations of G occurring in L2(GK) become heavily concentrated near certain submanifolds of GK as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators = PλqPλ, where Pλ : L2(GK) → Vλ is an orthogonal projection onto a G-irreducible summand, and q: G/KR is a continuous function acting on L2(GK) by multiplication.  相似文献   

9.
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: V(x) = O(¦ x ¦?(12)), grad V(x) = O(¦ x ¦?(32)), ΛV(x) = O(¦ x s?) (δ > 0), Λ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator F from PL2(R3) onto L2((0, ∞); L2(Ω)), P being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),
(α(H)(Pf,g)=0 (α(λ)(Ff)(λ),(Fg)(λ))L2(ω) dλ
.  相似文献   

10.
For an open set Ω ? RN, 1 ? p ? ∞ and λ ∈ R+, let W?pλ(Ω) denote the Sobolev-Slobodetzkij space obtained by completing C0(Ω) in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators U, 1 ? p, q ? ∞ and a quasibounded domain Ω ? RN. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map W?pλ(Ω) λ Lq(Ω) exists and belongs to the given Banach ideal U: Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any x ? Ω to the boundary ?Ω tends to zero as O(¦ x ¦?l) for ¦ x ¦ → ∞, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ N, μ > λ S(U; p,q:N) and v > N/l · λD(U;p,q), one has that W?pλ(Ω) λ Lq(Ω) belongs to the Banach ideal U. Here λD(U;p,q;N)∈R+ and λS(U;p,q;N)∈R+ are the D-limit order and S-limit order of the ideal U, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpnlqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in L2(Ω), where Ω fulfills condition C1l.For an open set Ω in RN, let W?pλ(Ω) denote the Sobolev-Slobodetzkij space obtained by completing C0(Ω) in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in RN and give sufficient conditions on λ such that the Sobolev imbedding operator W?pλ(Ω) λ Lq(Ω) exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in L2(Ω), where Ω is a quasibounded open set in RN.  相似文献   

11.
We consider two Gaussian measures P1 and P2 on (C(G), B) with zero expectations and covariance functions R1(x, y) and R2(x, y) respectively, where Rν(x, y) is the Green's function of the Dirichlet problem for some uniformly strongly elliptic differential operator A(ν) of order 2m, m ≥ [d2] + 1, on a bounded domain G in Rd (ν = 1, 2). It is shown that if the order of A(2) ? A(1) is at most 2m ? [d2] ? 1, then P1 and P2 are equivalent, while if the order is greater than 2m ? [d2] ? 1, then P1 and P2 are not always equivalent.  相似文献   

12.
A formula for the resolvent R(λ, T) of a Baxter operator T, on a complex Banach algebra A with identity e, is obtained. With the parameter θ ≠ 0 and e, but under some restriction, this formula is analogous to that for the resolvent of an averaging operator. A counterexample is given, which shows that such a Baxter operator is not averaging in general. When θ is regular in A, a simple representation of T in terms of summation and averaging operators is obtained.  相似文献   

13.
We regard a graph G as a set {1,…, v} together with a nonempty set E of two-element subsets of {1,…, v}. Let p = (p1,…, pv) be an element of Rnv representing v points in Rn and consider the realization G(p) of G in Rn consisting of the line segments [pi, pj] in Rn for {i, j} ?E. The figure G(p) is said to be rigid in Rn if every continuous path in Rnv, beginning at p and preserving the edge lengths of G(p), terminates at a point q ? Rnv which is the image (Tp1,…, Tpv) of p under an isometry T of Rn. We here study the rigidity and infinitesimal rigidity of graphs, surfaces, and more general structures. A graph theoretic method for determining the rigidity of graphs in R2 is discussed, followed by an examination of the rigidity of convex polyhedral surfaces in R3.  相似文献   

14.
In this paper we discuss the problem of determining a T-periodic solution x1(·, λ) of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space Rk. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?0 > 0 such that the differential equation has a T-periodic solution x1(·, λ(?)) for all ? satisfying 0 < ? < ?0. More specifically it is usually assumed that λ(?) has the form λ(?) = 0 where λ0 is a fixed vector in Rk. This means that attention is confined in the perturbation procedure to examining the dependence of x1(·, λ) on λ as λ varies along a line segment terminating at the origin in the parameter-space Rk. The results established here generalize this previous work by allowing one to study the dependence of x1(·, λ) on λ as λ varies through a “conical-horn” whose vertex rests at the origin in Rk. In the process an implicit-function formula is developed which is of some interest in its own right.  相似文献   

15.
Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator N is defined by Nf(x) = ?lim∈←0{E[f(Uξ))] ? f(x)}/Eξ, where τx? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that Nf(x) = ?trace D2f(x)+〈Df(x),x〉 for a large class of functions f. Let rt(x, dy) be the transition probabilities of U(t). The λ-potential Gλf, λ > 0, and normalized potential Rf of f are defined by Gλf(X) = ∫0e?λtrtf(x) dt and Rf(x) = ∫0 [rtf(x) ? rtf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D2Gλf(x) ? 〈DGλf(x), x〉 = ?f(x) + λGλf(x) and trace D2Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫Bf(y)p1(dy), where p1 is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.  相似文献   

16.
Nonlinear partial differential operators G: W1,p(Ω) → Lq(Ω) (1 ? p, q ∞) having the form G(u) = g(u, D1u,…, DNu), with g?C(R × RN), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, W1,∞(Ω), and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of W1,p(Ω) and Lq(Ω).  相似文献   

17.
A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C0-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)?1 exists and the family {(λ ? β)k(λI ? A)?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra Bг(X) which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has limkz (I ? k?1tA)?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C0-semigroup by an operator which is an element of Bг(X) is obtained.  相似文献   

18.
Let (i, H, E) and (j, K, F) be abstract Wiener spaces and let α be a reasonable norm on E ? F. We are interested in the following problem: is (i ? j, H \?bo2 K, E \?boαF) an abstract Wiener space ? The first thing we do is to prove that the setting of the problem is meaningfull: namely, i ? j is always a continuous one to one map from H \?bo2 K into E \?boαF. Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F=Lp(X,X,λ) for some σ-finite measure λ ? 0 then (i?j, H?2K,Lp(X,X,λ) is an abstract Wiener space. By-products are some new results on γ-radonifying operators, and new examples of Banach spaces and cross norms for which the answer is affirmative (in particular α = π the projective norm, and F=L1(X,X,λ)).  相似文献   

19.
Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by Cp(G)(0<p?2), (ii) the K-biinvariant elements in Cp(G) by Ip(G), (iii) the positive definite (zonal) spherical functions by P, and (iv) the spherical transform on Cp(G) by ? → \?gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto Cl(G), (ii) there exists a unique measure μ of polynomial growth on P such that T[ψ]=∫pψdμ for all ψ?I1(G) (iii) all measures μ of polynomial growth on P are obtained in this way, and (iv) T may be extended to a particular Ip(G) space (1 ? p ? 2) if and only if the support of μ lies in a certain easily defined subset of P. These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.  相似文献   

20.
The spectrum and essential spectrum in Lp(Rn) of a strongly Carleman pseudo-differential operator with symbol of class S?,0m, 0 ? p ? 1, are shown to coincide with the range of the symbol for some (but need not be all) p different from 2. The absolutely continuous spectra of perturbations of operators with strongly Carleman symbols (but without the assumption of being in S?,0m) are also investigated.  相似文献   

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