共查询到20条相似文献,搜索用时 234 毫秒
1.
We develop a systematic perturbation and resonance theory for the one-dimensional Schrödinger equation of the form $$( - d^2 /dx^2 + U(x) + \lambda V(x) - E)\psi (x) = 0,0 \leqq x< \infty ,$$ where the barrier potentialV(x) is supported only wherex≧1 and is non-negative there, and λ is a real parameter tending to infinity. We prove that every λ=∞ eigenvalue turns into a resonance or an eigenvalue for finite λ. 相似文献
2.
Stewart D. Johnson 《Communications in Mathematical Physics》1989,122(2):293-320
Families of unimodal maps satisfying
- T λ: [?1,1]?[?1,1] withT(±1)=?1 and |T λ ′ (1)|>1.
- T λ(x) isC 2 inx 2 and λ, and symmetric inx.
- T 0(0)=0,T 1(0)=1 with \(\frac{d}{{d\lambda }}\) T λ(0)>0
3.
David Gurarie 《Communications in Mathematical Physics》1987,112(3):491-502
We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??2+x 2?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)~|x|?α V(x) with a trigonometric even functionV(x)=Σa mcosω m x. The eigenvalues ofL are shown to be λ k =k+μ k with small μ k =O(k ?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ?α|V(x) ofB from asymptotics of {µ k }. 1 ∞ 相似文献
4.
We consider the initial value problem for the Zakharov equations $$\begin{gathered} \left( Z \right)\frac{1}{{\lambda ^2 }}n_{tt} - \Delta (n + \left| {\rm E} \right|^2 ) = 0n(x,0) = n_0 (x) \hfill \\ n_t (x,0) = n_1 (x) \hfill \\ iE_t + \Delta E - nE = 0E(x,0) = E_0 (x) \hfill \\ \end{gathered} $$ (x∈? k ,k=2, 3,t ≧0) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval independent of λ, a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as λ → ∞ to a solution of the cubic nonlinear Schrödinger equation (CSE)iE t +ΔE+|E|2 E=0. We consider both weak and strong solutions. For the case of strong solutions the results are analogous to previous results on the incompressible limit of compressible fluids. 相似文献
5.
Thomas Spencer 《Communications in Mathematical Physics》1975,44(2):143-164
We extend methods of high temperature expansions to show that for even weakly coupledP(?)2 quantum field models the Bethe-Salpeter kernel has 4 particle decay. More precisely ifK denotes the Euclidean Bethe-Salpeter kernel $$|K(x_1 ,x_2 ,x_3 ,x_4 )| \leqq Oe^{ - m_0 (1 - \varepsilon )d_2 } ,$$ wherex=(x 0,x 1),d 2=2|x 1 0 +x 2 0 ?x 3 0 ?x 4 0 |+|x 1 0 ?x 2 0 |+|x 3 0 ?x 4 0 | and ε(λ)→0 as λ→0. Our methods apply to otherr particle irreducible kernels. 相似文献
6.
Massimo Campanino Abel Klein J. Fernando Perez 《Communications in Mathematical Physics》1991,135(3):499-515
We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z d, σ1, σ3 are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ3(x)σ3(y)〉 and prove:
- Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z d withC x h <∞.
- Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z d.
7.
We consider magnetic Schrödinger operators $$H(\lambda \vec a) = ( - i\nabla - \lambda \vec a(x))^2$$ inL 2(R n ), where $\vec a \in C^1 (R^n ;R^n )$ and λεR. LettingM={x;B(x)=0}, whereB is the magnetic field associated with $\vec a$ , and $M_{\vec a} = \{ x;\vec a(x) = 0\}$ , we prove that $H(\lambda \vec a)$ converges to the (Dirichlet) Laplacian on the closed setM in the strong resolvent sense, as λ→∞,provided the set $M\backslash M_{\vec a}$ has measure zero. In various situations, which include the case of periodic fields, we even obtain norm resolvent convergence (again under the condition that $M\backslash M_{\vec a}$ has measure zero). As a consequence, if we are given a periodic fieldB where the regions withB=0 have non-empty interior and are enclosed by the region withB≠0, magnetic wells will be created when λ is large, opening up gaps in the spectrum of $H(\lambda \vec a)$ . We finally address the question of absolute continuity of $\vec a$ for periodic $H(\vec a)$ . 相似文献
8.
We establish existence of a dense set of non-linear eigenvalues,E, and exponentially localized eigenfunctions,u E , for some non-linear Schrödinger equations of the form $$Eu_E (x) = [( - \Delta + V(x))u_E ](x) + \lambda u_E (x)^3 ,$$ bifurcating off solutions of the linear equation with λ=0. The pointsx range over a lattice, ? d ,d=1,2,3,..., Δ is the finite difference Laplacian, andV(x) is a random potential. Such equations arise in localization theory and plasma physics. Our analysis is complicated by the circumstance that the linear operator ?Δ+V(x) has dense point spectrum near the edges of its spectrum which leads to small divisor problems. We solve these problems by developing some novel bifurcation techniques. Our methods extend to non-linear wave equations with random coefficients. 相似文献
9.
The purpose of this paper is to study the so-calledspectral class Q of anharmonic oscillatorsQ=?D 2+q having the same spectrum λ n =2n (n≧0) as the harmonic oscillatorQ 0=?D 2+x 2?1. Thenorming constants \(t_n = \mathop {\lim }\limits_{x \uparrow \infty } \ell g[( - 1)^n {{e_n (x)} \mathord{\left/ {\vphantom {{e_n (x)} {e_n }}} \right. \kern-0em} {e_n }}( - x)]\) of the eigenfunctions ofQ form a complete set of coordinates inQ in terms of which the potential may be expressed asq=x 2?1?2D 2 ?g? with $$\theta = \det \left[ {\delta _{ij} + (e^{ti} - 1)\int\limits_x^\infty {e_i^0 e_j^0 :0 \leqq i,j,< \infty } } \right],$$ e n 0 being then th eigenfunctionQ 0. The spectrum and norming constants are canonically conjugate relative to the bracket [F, G]=∫ΔFDΔGdx,to wit: [λ i , λj=0, [t i, 2λ j ]=1 or 0 according to whetheri=j or not, and [t i,t j]=0. This prompts an investigation of the symplectic geometry ofQ. The function ? is related to the theta function of a singular algebraic curve. Numerical results are also presented. 相似文献
10.
We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH (0) is diagonal in a basis ∣s〉=? x ∣s x 〉 which may be labeled by the configurationss={sx} of a suitable classical spin system on ? d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH (0)(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH (0), provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper. 相似文献
11.
K. R. Ito 《Communications in Mathematical Physics》1982,83(4):537-561
Let ν=detren(1+K g ) be the renormalized Matthews-Salam determinant of (QED)2, where \(K_g = ieA_{g,} S = \left( {\sum {\gamma _\mu \partial } _\mu + m} \right)^{ - 1} \) is euclidean fermion propagator of one of the following boundary conditions: (1) free, (2) periodic at ?Λ, Λ=[?L/2;L/2]2, (3) anti-periodic at ?Λ, and \(A_g (x) = (\sum \gamma _\mu A_\mu (x))g(x)\) . Hereg(x)=1 ifxεΛ0=[?r/2,r/2]2 с Λ and 0 otherwise. Then we show
- νεL p (dμ(A)), p>0. Further we prove a new determinant inequality which holds for the QED, QCD-type models containing fermions. This enables us to prove:
- Z(Λ0)=∫νdμ(A)≦exp[c|Λ0|]. Similar volume dependence is shown for the Schwinger functions.
12.
We consider the zero-temperature behavior of a disordered array of quantum rotators given by the finite-volume Hamiltonian: $$H_\Lambda = - \mathop \Sigma \limits_{x \in \Lambda } \frac{{h(x)}}{2}\frac{{\partial ^2 }}{{\partial \varphi (x)^2 }} - J\mathop \Sigma \limits_{\left\langle {x,y} \right\rangle \in \Lambda } \cos (\varphi (x) - \varphi (y))$$ , wherex,y∈Z d , 〈,〉 denotes nearest neighbors inZ d ;J>0 andh={h(x)>0,x∈Z d } are independent identically distributed random variables with common distributiondμ(h), satisfying ∫h ?δ dμ(h)<∞ for some δ>0. We prove that for anym>0 it is possible to chooseJ(m) sufficiently small such that, if 0<J<J(m), for almost every choice ofh and everyx∈Z d the ground state correlation function satisfies $$\left\langle {\cos (\varphi (x) - \varphi (y))} \right\rangle \leqq C_{x,h,J} e^{ - m\left| {x - y} \right|} $$ for ally∈Z d withC x,h,J <∞. 相似文献
13.
Steven Schochet 《Communications in Mathematical Physics》1985,102(2):207-215
The incompressible limit in nonlinear elasticity is shown to fall under the theory of singular limits of quasilinear symmetric hyperbolic systems developed by Klainerman and Majda. Specifically, initial-value problems for a family of hyperelastic materials with stored energy functions $$W\left( {\frac{{\partial x}}{{\partial X}}} \right) = W_\infty \left( {\frac{{\partial x}}{{\partial X}}} \right) + \lambda ^2 w\left( {\det \frac{{\partial x}}{{\partial X}}} \right)$$ are considered, whereX andx are reference and deformed coordinates respectively. Under the assumption that the elasticity tensor $$A_{kl}^{ij} \equiv \frac{{\partial ^2 W_\infty }}{{\partial \left( {\frac{{\partial x^i }}{{\partial X^k }}} \right)\partial \left( {\frac{{\partial x^j }}{{\partial X^l }}} \right)}}$$ is positive definite near the identity matrix and thatw″(1)>0, the following results are proven for appropriate initial data: i) existence of solutions of the corresponding evolution equations on a time interval independent of λ as λ→∞, and ii) convergence as λ → ∞ of the solutions to a solution of the incompressible elastodynamics equations. 相似文献
14.
V. R. Zoller 《Zeitschrift fur Physik C Particles and Fields》1992,54(3):425-429
The perturbative QCD is applied to calculation of nuclear shadowing correction to the deuterium structure functionF 2 D (x) atx«1. This correction is shown to be numerically large. It is emphasized that the neglect of nuclear shadowing leads to underestimating of the neutron structure functionF 2 n (x) and, as a result, to gross overestimating of the Gottfried integral \(\int_0^1 {\left[ {F_2^P (x) - F_2^n (x)} \right]} dx/x\) 相似文献
15.
16.
In this paper we will study the nonlinear Schrödinger equations: $$\begin{gathered} i\partial _t u + \tfrac{1}{2}\Delta u = \left| u \right|^2 u, (t,x) \in \mathbb{R} \times \mathbb{R}_x^n , \hfill \\ u(0,x) = \phi (x), x \in \mathbb{R}_x^n \hfill \\ \end{gathered} $$ . It is shown that the solutions of (*) exist and are analytic in space variables fort∈??{0} if φ(x) (∈H 2n+1,2(? x n )) decay exponentially as |x|→∞ andn≧2. 相似文献
17.
We consider the class of quasi-periodic self-adjoint operators?(x)) = \(\hat D(x) + \hat V(x)\) ,x∈S 1=?1/?1, on a multi-dimensional lattice ? v , with the matrix elements $$\hat D_{mn} (x) = \delta _{mn} D(x + n\omega ), \hat V_{mn} (x) = V(m - n, x + n\omega )$$ , whereD(x+1) =D(x), V(n, s+1) =V(n, x), ω ∈ ? v and |V(n, x)| ≤ εe ?r|n|,r > 0. We prove that, if ε is small enough,V(n,·) andD(·) satisfy some conditions of smoothness, andD(·) is non-degenerate, then for a.e. ω and for a.e.x∈S 1 the operator?(x) has pure point spectrum. All its eigenfunctions belong tol 1(? v ). 相似文献
18.
LetS ?=??Δ+V, withV smooth. If 0<E 2V(x), the spectrum ofS ? nearE 2 consists (for ? small) of finitely-many eigenvalues,λ j (?). We study the asymptotic distribution of these eigenvalues aboutE 2 as ?→0; we obtain semi-classical asymptotics for $$\sum\limits_j {f\left( {\frac{{\sqrt {\lambda _j (\hbar )} - E}}{\hbar }} \right)} $$ with \(\hat f \in C_0^\infty \) , in terms of the periodic classical trajectories on the energy surface \(B_E = \left\{ {\left| \xi \right|^2 + V(x) = E^2 } \right\}\) . This in turn gives Weyl-type estimates for the counting function \(\# \left\{ {j;\left| {\sqrt {\lambda _j (\hbar )} - E} \right| \leqq c\hbar } \right\}\) . We make a detailed analysis of the case when the flow onB E is periodic. 相似文献
19.
Schrödinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist
We consider the Schrödinger-like operatorH in which the role of a potential is played by the lattice sum of rank 1 operators \(|\left. {v_n } \right\rangle \left\langle {v_n |} \right.\) multiplied by g tan π[(α,n)+ω],g>0, α∈? d ,n∈? d , ω∈[0, 1]. We show that if the vector α satisfies the Diophantine condition and the Fourier transform support of the functionsv n (x)=v(x-n),x∈? d ,n∈? d , is small then the spectrum ofH consists of a dense point component coinciding with? and an absolutely continuous component coinciding with [?, ∞), where ? is the radius of the mentioned support. Besides, we find the integrated density of statesN(λ) (it has a jump at λ=?) and zero temperature a.c. conductivityσ λ (v), that also has a jump at λ=? and vanishes faster than any power of the external field frequency ν as ν→0 and λ≠?. 相似文献
20.
Kenji Yajima 《Communications in Mathematical Physics》1996,181(3):605-629
The fundamental solutionE(t,s,x,y) of time dependent Schrödinger equationsi?u/?t=?(1/2)Δu+V(t,x)u is studied. It is shown that ?E(t,s,x,y) is smooth and bounded fort≠s if the potential is sub-quadratic in the sense thatV(t,x)=o(|x|2) at infinity; ? in one dimension, ifV(t,x)=V(x) is time independent and super-quadratic in the sense thatV(x)≧C(1+|x|)2+ε at infinity,C>0 and ε>0, thenE(t,s,x,y) is nowhereC 1. The result is explained in terms of the limiting behavior as the energy tends to infinity of the corresponding classical particle. 相似文献