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1.
When solving the inverse scattering problem for a discrete Sturm–Liouville operator with a rapidly decreasing potential,
one gets reflection coefficients s
± and invertible operators , where is the Hankel operator related to the symbol s
±. The Marchenko–Faddeev theorem [8] (in the continuous case, for the discrete case see [4, 6]), guarantees the uniqueness
of the solution of the inverse scattering problem. In this article we ask the following natural question – can one find a
precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators are invertible? Can one claim that uniqueness implies invertibility or vise versa?
Moreover, we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost
periodic potentials. So we merge here two mostly developed cases of the inverse problem for Sturm–Liouville operators: the
inverse problem with (almost) periodic potential and the inverse problem with the fast decreasing potential.
Received: 7 September 2001 / Accepted: 3 December 2001 相似文献
2.
We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The
magnetic field is generated by a scalar potential hence we bypass the usual A∈L
2
loc condition on the vector potential, which does not allow to consider such singular fields. We extend the Aharonov–Casher
theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite
total variation. One of the key technical tools is a weighted L
2 estimate on a singular integral operator.
Received: 14 May 2001 / Accepted: 5 September 2001 相似文献
3.
Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based
on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which
admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure
theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, the Kolmogorov
sum rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple
examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result
for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms
of quantum covers. 相似文献
4.
We investigate the conditions under which the class of ghost-free, tachyon-freeR + R
2 +Q
2 theories with torsion satisfy Birkhoff's theorem. We prove a weakened Birkhoff theorem requiring an additional assumption of parity invariance for two Lagrangians one of which contains torsion squared terms in addition to curvature squared terms. For another Lagrangian, also containing torsion squared terms, a weakened Birkhoff theorem requiring the additional assumptions of parity invariance and constant scalar curvature is proven. A special case of this Lagrangian is shown to satisfy a weakened Birkhoff theorem requiring only the additional assumption of constant scalar curvature. In addition the explicit dependence of torsion on parity noninvariant quantities is displayed. 相似文献
5.
In terms of beable algebras Halvorson and Clifton [International Journal of Theoretical Physics 38 (1999) 2441–2484] generalized the uniqueness theorem (Studies in History and Philosophy of Modern Physics 27 (1996) 181–219] which characterizes interpretations of quantum mechanics by preferred observables. We examine whether dispersion-free
states on beable algebras in the generalized uniqueness theorem can be regarded as truth-value assignments in the case where
a preferred observable is the set of all spectral projections of a density operator, and in the case where a preferred observable
is the set of all spectral projections of the position operator as well. 相似文献
6.
Han Dong Ying-bin Wang Xin-he Meng 《The European Physical Journal C - Particles and Fields》2012,72(5):2002
f(T) theory, a generally modified teleparallel gravity, has been proposed as an alternative gravity model to account for the
dark energy phenomena. Following our previous work [Xin-he Meng and Ying-bin Wang, Eur. Phys. J. (2011), ], we prove that Birkhoff’s theorem holds in a more general context, specifically with the off diagonal tetrad case, in this
communication letter. Then, we discuss, respectively, the results of the external vacuum and internal gravitational field
in the f(T) gravity framework, as well as the extended meaning of this theorem. We also investigate the validity of Birkhoff’s theorem
in the frame of f(T) gravity via a conformal transformation by regarding the Brans–Dicke-like scalar as effective matter, and study the equivalence
between both Einstein frame and Jordan frame. 相似文献
7.
F. Ehlotzky 《Il Nuovo Cimento D》1994,16(5):453-462
Summary We show that for potential scattering of electrons in a bichromatic laser field a Kroll-Watson type of theorem can be derived
in which the scatteringT-matrix element is composed of a renormalized on-shell matrix element for scattering without a field times a generalized Bessel
function factor. The radiation field has two components of frequency ω1 and sω1 (s=2,3,…) and both components are out of phase by an angle ϕ. Our paper is a generalization of earlier investigations which
were performed in the first-order Born approximation. 相似文献
8.
Yuri Bakhtin 《Journal of statistical physics》2006,122(2):351-360
We consider the 3D Navier–Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic
process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness
theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following “one force—one solution” principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation. 相似文献
9.
The class of the even-power series potentials,V(r)=-D+∑
k-0
∞
Vkλkr2k+2,V
0=ω2>0is studied with the aim of obtaining approximate analytic expressions for the nonrelativistic energy eigenvalues, the expectation
values for the potential and kinetic energy operators, and the mean square radii of the orbits of a particle in its ground
and excited states. We use the hypervirial theorems (HVT) in conjunction with the Hellmann-Feynman theorem (HFT), which provide
a very powerful scheme for the treatment of the above and other types of potentials, as previous studies have shown. The formalism
is reviewed and the expressions of the above-mentioned quantities are subsequently given in a convenient way in terms of the
potential parameters, the mass of the particle, and the corresponding quantum numbers, and are then applied to the case of
the Gaussian potential and to the potentialV(r)=−D/cosh2(r/R). These expressions are given in the form of series expansions, the first terms of which yield, in quite a number of cases,
values of very satisfactory accuracy. 相似文献
10.
V. N. Tolstoy 《Czechoslovak Journal of Physics》2001,51(12):1453-1458
Tensor operators are discussed for Hopf algebras and, in particular, for a quantum (q-deformed) algebraUq(g), whereg is any simple finite-dimensional or affine Lie algebra. These operators are defined via an adjoint action in a Hopf algebra.
There are two types of the tensor operators which correspond to two coproducts in the Hopf algebra. In the case of tensor
products of two tensor operators one can obtain 8 types of the tensor operators and so on. We prove the relations which can
be a basis for a proof of the Wigner-Eckart theorem for the Hopf algebras. It is also shown that in the case ofUq(g) a scalar operator can be differed from an invariant operator but atq=1 these operators coincide.
Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June
2001.
Supported by Russian Foundation for Fundamental Research, grant 99-01-01163, and by INTAS-00-00055. 相似文献
11.
Lajos Molnár 《Communications in Mathematical Physics》2001,217(2):409-421
Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated,
for example, in the following way: Every bijective transformation on the set ℒ of all 1-dimensional subspaces of a Hilbert
space H which preserves the angle between the elements of ℒ is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n∈ℕ fixed.
Received: 28 August 2000 / Accepted: 30 October 2000 相似文献
12.
We study the O(N) symmetric linear sigma-model at finite temperature as the low-energy effective models of quantum chromodynamics (QCD) using
the Cornwall-Jackiw-Tomboulis (CJT) effective action for composite operators. It has so far been claimed that the Nambu-Goldstone
theorem is not satisfied at finite temperature in this framework unless the large-N limit in the O(N) symmetry is taken. We show that this is not the case. The pion is always massless below the critical temperature, if one
determines the propagator within the form such that the symmetry of the system is conserved, and defines the pion mass as
the curvature of the effective potential. We use a regularization for the CJT effective potential in the Hartree approximation,
which is analogous to the renormalization of auxiliary fields. A numerical study of the Schwinger-Dyson equation and the gap
equation is carried out including the thermal and quantum loops. We point out a problem in the derivation of the sigma meson
mass without quantum correction at finite temperature. A problem about the order of the phase transition in this approach
is also discussed.
Received: 21 June 2000 / Accepted: 13 September 2000 相似文献
13.
We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group
associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms
a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants
is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine
their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their
classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem
of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical
case with orthogonal quantum group symmetry. 相似文献
14.
Florian Theil 《Communications in Mathematical Physics》2006,262(1):209-236
Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be
studied are pair interaction energies of the type where y(x) ∈ℝ2 is the position of particle x and V(r) ∈ ℝ is the pair-interaction energy of two particles which are placed at distance r. Due to the Mermin-Wagner theorem it can't be expected that at finite temperature this system exhibits long-range ordering.
We focus on the zero temperature case and show rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges
to an explicit constant E*: where E* ∈ ℝ is the minimum of a simple function on [0,∞). Furthermore, if suitable Dirichlet- or periodic boundary conditions are
used, then the minimizers form a triangular lattice. To the best knowledge of the author this is the first result in the literature
where periodicity of ground states is established for a physically relevant model which is invariant under the Euclidean symmetry
group consisting of rotations and translations. 相似文献
15.
Bart D’Hooghe 《International Journal of Theoretical Physics》2010,49(12):3069-3084
We adopt an operational approach to quantum mechanics in which a physical system is defined by the mathematical structure
of its set of states and properties. We present a model in which the maximal change of state of the system due to interaction
with the measurement context is controlled by a parameter which corresponds with the number N of possible outcomes in an experiment. In the case N=2 the system reduces to a model for the spin measurements on a quantum spin-1/2 particle. In the limit N→∞ the system is classical, i.e. the experiments are deterministic and its set of properties is a Boolean lattice. For intermediate
situations the change of state due to measurement is neither ‘maximal’ (i.e. quantum) nor ‘zero’ (i.e. classical). We show
that two of the axioms used in Piron’s representation theorem for quantum mechanics are violated, namely the covering law
and weak modularity. Next, we discuss a modified version of the model for which it is even impossible to define an orthocomplementation
on the set of properties. Another interesting feature for the intermediate situations of this model is that the probability
of a state transition in general not only depends on the two states involved, but also on the measurement context which induces
the state transition. 相似文献
16.
Davood Momeni 《International Journal of Theoretical Physics》2011,50(5):1493-1514
Hořava proposed a non-relativistic renormalizable theory of gravitation, which is reduced to general relativity (GR) in large
distances (infra-red regime (IR)). It is believed that this theory is an ultra-violet (UV) completion for the classical theory
of gravitation. In this paper, after a brief review of some fundamental features of this theory, we investigate it for a static
cylindrical symmetric solution which describes Cosmic string as a special case. We have also investigated some possible solutions, and have seen that how the classical GR field equations
are modified for generic potential V(g). In one case there is an algebraic constraint on the values of three coupling constants. Finally as a pioneering work we
deduce the most general cosmic string in this theory. We explicitly show that how the coupling constants distort the mass parameter of cosmic string. We deduce an explicit function for mass per unit length of the space-time as a function of the coupling constants. We compare this function with another which Aryal et al. (Phys. Rev. D 34:2263, 1986) have found in GR. Also we calculate the self-force on a massive particle near Hořava-Lifshitz straight string and we give
a typical order for the coupling constant
g
9. This order of magnitude proposes a cosmological test for validity of this theory. 相似文献
17.
We study the Dirac operator D
0 in an external potential V, coupled to a quantized radiation field with energy H
f and vector potential A. Our result is a Chernoff-type theorem, i.e., we prove, for the operator D
0+α · A+V +λ H
f with λ ∈{0, 1}, that the essential self-adjointness is not affected by the behavior of V at ∞.
相似文献
18.
We present a new solution for the rotation curves of galactic disks with gravitational potential of the Yukawa type. We follow
the technique employed by Toomre in 1963 in the study of galactic disks in the Newtonian theory. This new solution allows
an easy comparison between the Newtonian solution and the Yukawian one. Therefore, constraints on the parameters of theories
of gravitation can be imposed, which in the weak field limit reduce to Yukawian potentials. We then apply our formulae to
the study of rotation curves for a zero-thickness exponential disk and compare it with the Newtonian case studied by Freeman
in 1970. As an application of the mathematical tool developed here, we show that in any theory of gravity with a massive graviton
(this means a gravitational potential of the Yukawa type), a strong limit can be imposed on the mass (m
g) of this particle. For example, in order to obtain a galactic disk with a scale length of b∼ 10 kpc, we should have a massive graviton of m
g ≪ 10−59g. This result is much more restrictive than those inferred from solar system observations. 相似文献
19.
M. V. Ivanchenko 《JETP Letters》2010,92(6):365-369
Nonlinearity shapes lattice dynamics affecting vibrational spectrum, transport and thermalization phenomena. Beside breathers
and solitons one finds the third fundamental class of nonlinear modes, q-breathers, i.e., periodic orbits in nonlinear lattices, exponentially localized in the reciprocal mode space. To date, the
studies of q-breathers have been confined to the cubic and quartic nonlinearity in the interaction potential. In this paper we study the
case of arbitrary nonlinearity index γ in an acoustic chain. We uncover qualitative difference in the scaling of delocalization
and stability thresholds of q-breathers with the system size: there exists a critical index γ* = 6, below which both thresholds (in nonlinearity strength)
tend to zero, and diverge when above. We also demonstrate that this critical index value is decisive for the presence or absence
of thermalization. For a generic interaction potential the mode space localized dynamics is determined only by the three lowest
order nonlinear terms in the power series expansion. 相似文献
20.
We formalize a classification of pair interactions based on the convergence properties of the forces acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function
(PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e.,
in the “usual” thermodynamic limit. For a pair interaction potential V(r) with V(r→∞)∼1/r
γ
defining a bounded pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the pair force is absolutely integrable, i.e., for γ>d−1, where d is the spatial dimension. We refer to this case as dynamically short-range, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood
around it. For the dynamically long-range case, i.e., γ≤d−1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined
in this limit. We discuss also how, for γ≤d−1 (and notably, for the case of gravity, γ=d−2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization
of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant
classification in this context is, however, that which divides pair forces with γ>d−2 (or γ<d−2), for which the PDF of the difference in forces is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as
for the (marginal) case of gravity, be defined consistently in an infinite uniform system. 相似文献