共查询到20条相似文献,搜索用时 93 毫秒
1.
We explore connections between Krein's spectral shift function ζ(λ,H
0, H) associated with the pair of self-adjoint operators (H
0, H),H=H
0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K
*(H
0−λ−i0)−1
K) associated with the operator-valued Herglotz functionJ+K
*(H
0−z)−1
K, Im(z)>0 inH, whereV=KJK
* andJ=sgn(V). Our principal results include a new representation for ζ(λ,H
0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E
J+A(λ)+tB(λ)(−∞, 0)),E
J((−∞, 0))), ℝ, whereA(λ)=Re(K
*(H
0−λ−i0−1
K),B(λ)=Im(K
*(H
0−λ-i0)−1
K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H
0, H) coincides with the trindex associated with the pair (Ξ(J+K
*(H
0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm
determinant of the abstract scattering matrix.
We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting
functions by appropriate spectral shift functions. 相似文献
2.
Zhu Fuliu 《数学学报(英文版)》1997,13(4):545-552
LetG/K be the noncompact Riemannian symmetric spaceSL(3,H)/Sp(3). We shall prove in this paper that forf∈L
p(SL(3,H)/Sp(3)), 1≤p≤2, the Riesz means of orderz off with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge tof for Rez >δ(n,p). The critical index δ(n,p) is the same as in the classical Stein's result for Euclidean space, and as in the noncompact symmetric spaces of rank one
and of complex type.
Partially supported by National Natural Science Foundation of China 相似文献
3.
Yosef Stein 《Israel Journal of Mathematics》1989,68(1):109-122
LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏
i=1
n(λ)
A
iλ
k
μ whereA
iλ are distinct primes. Let ϱλ(A) =n(λ) − 1 and let ρ(A) = Σλɛσ(A)ϱλ(A). The main result is the following:
Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA. 相似文献
4.
Michel Bonnefont 《Potential Analysis》2012,36(2):275-300
In this paper, we study a subelliptic heat kernel on the Lie group SL(2, ℝ) and on its universal covering
[(SL(2,\mathbbR))\tilde]\widetilde{\mathbf{SL}(2,\mathbb{R})}. The subelliptic structure on SL(2,ℝ) comes from the fibration SO(2)→SL(2,ℝ) →H
2 and it can be lifted to
[(SL(2,\mathbbR))\tilde]\widetilde{\mathbf{SL}(2,\mathbb{R})}. First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics
in small times of the heat kernels and give us a way to compute the subriemannian distance. Then, we establish some gradient
estimates and some functional inequalities like a Li-Yau type estimate and a reverse Poincaré inequality that are valid for
both heat kernels. 相似文献
5.
Fang Liping 《数学学报(英文版)》1998,14(1):139-144
Letf be a holomorphic self-map of the punctured plane ℂ*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI
0(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞} andI
∞(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞}. We try to find the relation betweenI
0(f),I
∞(t) andJ(f). It is proved that both the boundary ofI
0(f) and the boundary ofI
∞)f) equal toJ(f),I
0(f) ∩J(f) ≠ θ andI
∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI
0(f) andI
∞(f) are not doubly-bounded.
Supported by the National Natural Science Foundation of China 相似文献
6.
Dominique Arlettaz 《Central European Journal of Mathematics》2004,2(1):50-56
This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism
b
n
: π
n
(X)→H
n
(H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π
n
(X)→E
n
(X), F
n
(X)→(E∧F)
n
(X), F
n
(X)→H
n
(X;π
0
F) and F
n
(X)→H
n+t
(X;π
t
F) for other cohomology theories E
*(−) and F
*(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in
terms of the order of the Postnikov invariants of the spectrum F. 相似文献
7.
Laura DeMarco 《Mathematische Annalen》2003,326(1):43-73
Let L(f)=∫log∥Df∥dμ
f
denote the Lyapunov exponent of a rational map, f:P
1→P
1
. In this paper, we show that for any holomorphic family of rational maps {f
λ
:λX} of degree d>1, T(f)=dd
c
L(f
λ
) defines a natural, positive (1,1)-current on X supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula
for the Lyapunov exponent:
Here F:C
2
→C
2
is a homogeneous polynomial lift of f; ; G
F
is the escape rate function of F; and capK
F
is the homogeneous capacity of the filled Julia set of F. We show, in particular, that the capacity of K
F
is given explicitly by the formula
where Res(F) is the resultant of the polynomial coordinate functions of F.
We introduce the homogeneous capacity of compact, circled and pseudoconvex sets K⊂C
2 and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such K⊂C
2 correspond to metrics of non-negative curvature on P
1, and we obtain a variational characterization of curvature.
Received: 28 November 2001 / Revised version: 2 April 2002 /
Published online: 10 February 2003 相似文献
8.
T. Shibata 《Annali di Matematica Pura ed Applicata》2007,186(3):525-537
We consider the nonlinear Sturm–Liouville problem
where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R
+ × L
2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction u
α with ‖ u
α ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ≔ f(u)/u
p
satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ∼ α
p−1
h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by lim
u → ∞
uh′(u).
Mathematics Subject Classification (2000) 34B15 相似文献
(1) |
9.
Let M(σ) = sup{|F(σ + it)|: t ∈ ℝ} and μ(σ) = max {|a
n
|exp(σλn): n ≥ 0}, σ < 0, for a Dirichlet series {fx995-01} with abscissa of absolute convergence σa = 0. We prove that the condition ln ln n = o(ln λn), n → ∞, is necessary and sufficient for the equivalence of the relations {fx995-02}, for each series of this type.
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 851–856, June, 2008. 相似文献
10.
Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL1(A). For a field extension K of F, we study the first Galois Cohomology group H 1(K,SL1(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series. 相似文献
11.
Eli Aljadeff 《Israel Journal of Mathematics》1994,86(1-3):221-232
LetK be a commutative ring with a unit element 1. Let Γ be a finite group acting onK via a mapt: Γ→Aut(K). For every subgroupH≤Γ define tr
H
:K→K
H
by tr
h
(x)=Σσ∈H
σ(x). We proveTheorem: trΓ
is surjective onto
K
Γ
if and only if tr
P
is surjective onto K
P
for every (cyclic) prime order subgroup P of Γ.
This is false for certain non-commutative ringsK. 相似文献
12.
Tom Cheatham 《Israel Journal of Mathematics》1979,33(2):172-176
R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom
R
(G, M) there exists μ∈Hom
R
(G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover. 相似文献
13.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
14.
Michel Talagrand 《Probability Theory and Related Fields》2001,121(2):237-268
We study the Hopfield model at temperature 1, when thenumber M(N) of patterns grows a bit slower than N. We reach a goodunderstanding of the model whenever M(N)≤N/(log N)11. For example, we show that if M(N)→∞, for two typical configurations σ
1, σ
2, (∑
i
≤
N
σ1
i
σ2
i
)2 is close to NM(N).
Received: 15 December 1999 / Revised version: 8 December 2000 / Published online: 23 August 2001 相似文献
15.
Marshall A. Whittlesey 《Arkiv f?r Matematik》1999,37(2):409-423
Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and letA(Δ) be the algebra of complex functions continuous on Δ and analytic in int Δ. LetK be a compact set in C2 such that Π(K)=Γ, and letK
λ≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,K
λ is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))2-S(λ) quadratic in w withR,S∈A(Δ) such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ intK
λ, whereS has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component
ofK
λ. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull ofK. Furthermore, we show that БК/K is the disjoint union of such disks. 相似文献
16.
On the Isolated Points of the Spectrum of Paranormal Operators 总被引:1,自引:0,他引:1
Atsushi Uchiyama 《Integral Equations and Operator Theory》2006,55(1):145-151
For paranormal operator T on a separable complex Hilbert space
we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π00(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)*. 相似文献
17.
Baruch Solel 《Israel Journal of Mathematics》1988,62(1):63-89
LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH
∞(α) be the associated analytic subalgebra; i.e.H
∞(α)={X ∈M: sp∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH
∞(α) isH
∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M
α = Ci)H
∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH
∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α). 相似文献
18.
For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T
*(J
(r,s,q)(Y, R
1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T
*(J
(r,s)
(Y, R)0)*→R for any Y as above. 相似文献
19.
E. Ballico 《Annali dell'Universita di Ferrara》1999,45(1):123-125
Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic
k
(X) the uniqueg
k
1
onX. SetL:=K
X⊗(R
*)⊗t.L is very ample. Leth
L:X→P(H
0(X, L)*) be the associated embedding. Here we prove thath
L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh
L(X) is scheme-theoretically cut out by quadrics.
The author was partially supported by MURST and GNSAGA of CNR (Italy). 相似文献
20.
Let H
1, H
2 be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H
1 and taking values in H
2. In this article we prove the following results:
We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results. 相似文献
(i) | Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T*T) of T*T, In addition, if H 1 = H 2 and T is self-adjoint, then |
(ii) | inf {‖T x‖: x ∈ D(T) ∩ N(T)⊥‖x‖ = 1} = inf {|λ|: 0 ≠ λ ∈ σ(T)} |
(iii) | Every isolated spectral value of T is an eigenvalue of T |
(iv) | Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T) of T |
(v) | σ(T) bounded implies T is bounded. |