Inequalities for traces on von Neumann algebras

A number of useful inequalities, which are known for the trace on a separable Hilbert space, are extended to traces on von Neumann algebras. In particular, we prove the Golden rule, Hölder inequality, and some convexity statements.Battelle Fellow, 1970–1971.     

Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces

LetM be a complete Riemannian surface with constant curvature –1, infinite volume, and a finitely generated fundamental group. Denote by (M) the lowest eigenvalue of the Laplacian onM, and let _M be the associated eigenfunction. We estimate the size of (M) and the shape of _M by a finite procedure which has an electrical circuit analogue. Using the Margulis lemma, we decomposeM into its thick and thin parts. On the compact thick components, we show that _M varies from a constant value by no more thanO( ). The estimate for (M) is calculable in terms of the topology ofM and the lengths of short geodesics ofM. An analogous theorem of the compact case was treated in [SWY].     

New Inequalities for Quantum Von Neumann and Tomographic Mutual Information

We study the entropic inequalities related to the quantum mutual information for bipartite system and tomographic mutual information for the Werner state of two qubits. We discuss quantum correlations corresponding to the entanglement properties of the qubits in the Werner state.     

The convergence of the lowest eigenvalue of a real nearly-symmetric matrix

The lowest eigenvalue of a real nearly-symmetric matrix is expressed as a perturbation series in terms of the eigenvalues of the symmetric part and the matrix elements of the skew-symmetric part. It is shown that the resulting series is closely related to the perturbation series for the lowest eigenvalue of a related hermitian matrix. This enables the behaviour of the lowest eigenvalue of a nearly symmetric matrix as the dimension of the matrix is increased to be deduced from the behaviour of the lowest eigenvalue of a hermitian matrix. This is of considerable importance as the behaviour of the lowest eigenvalue of a hermitian matrix as the dimension of the matrix is increased can be much more readily established. A possible application to Boys' transcorrelated method of calculating atomic and molecular energies is suggested.     

Strong-electric-field eigenvalue asymptotics for the perturbed magnetic Schrödinger operator

We consider the Schrödinger operator with constant full-rank magnetic field, perturbed by an electric potential which decays at infinity, and has a constant sign. We study the asymptotic behaviour for large values of the electric-field coupling constant of the eigenvalues situated in the gaps of the essential spectrum of the unperturbed operator.Partly supported by the Bulgarian Science Foundation under contract MM 33/91     

The magnetic moment of the lowest 3/2− state in57Co

The reaction⁵⁴Fe(α, n)⁵⁷Ni has been used to implant⁵⁷Co isotopes in ferromagnetic iron. Theg-factor of the lowest 3/2^? state is determined using the internal field in a constant angle reversed field method. The angular correlation of the 127–1,378 keV cascade is also measured. The result of the angular correlation measurement together with reaction data is consistent withp _3/2 andp _1/2 single particle assignments to the lowest 3/2^? resp. 1/2^? state. In view of this statement the quenching of the magnetic moment is discussed.     

厄米本征值问题的探究

给出了用探索性方法进行数学物理方法教学的一个案例.从厄米本征值问题出发,经过合情推理,归纳出厄米多项式的递推公式,并猜想出通项公式.该方法可以在传授知识的同时,培养学生的探索意识与创新能力.     

Iterative approach for the eigenvalue problems

An approximation method based on the iterative technique is developed within the framework of linear delta expansion (LDE) technique for the eigenvalues and eigenfunctions of the one-dimensional and three-dimensional realistic physical problems. This technique allows us to obtain the coefficient in the perturbation series for the eigenfunctions and the eigenvalues directly by knowing the eigenfunctions and the eigenvalues of the unperturbed problems in quantum mechanics. Examples are presented to support this. Hence, the LDE technique can be used for nonperturbative as well as perturbative systems to find approximate solutions of eigenvalue problems.     

All Inequalities for the Relative Entropy

The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party states to a smaller number m of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy.Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by quantum relative entropies. In doing so we make a connection to secret sharing schemes with general access structures: indeed, it turns out that the extremal rays of the cone defined by monotonicity are populated by classical secret sharing schemes.A surprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.     

Inequalities for the Schattenp-norm. III

We present some inequalities for the Schattenp-norm of operators on a Hilbert space. It is shown, among other things, that ifA is an operator such that ReAa0, then for any operatorX, AX+XA* _p 2aX _p . Also, for any two operatorsA andB, A–B ₂ ² +A*B* ₂ ² 2A–B ₂ ² .     

Inequalities for the Schattenp-norm. IV

We prove some inequalities for the Schattenp-norm of operators on a Hilbert space. It is shown, among other things, that ifA,B, andX are operators such thatA +B ≧ |X| andA +B ≧ |X*|, then ∥AX +XB∥ _p ^p + ∥AX* +X*B∥ _p ^p ≧2 ∥X∥ _2p ^2p for 1 ≦p<∞, and max (∥AX +XB∥, ∥AX* +X*B∥) ≧ ∥X∥². Also, for any three operatorsA,B, andX, $$|| |A|X - X|B| ||_2^2 + || |A*|X - X|B*| ||_2^2 \leqq ||AX - XB||_2^2 + ||A*X - XB*||_2^2 .$$      

An asymptotic expansion for the Neumann sieve problem

In this paper, we study the Neumann sieve problem for the Laplace equation. Our objective is to compute the complete asymptotic expansion for the problem. The expansion consists of the interior part, in the vicinity of the filter, and an exterior part, far away from the filter. The interior approximation is a Bakhvalov-Panasenko-type expansion with terms defined by a sequence of auxiliary problems on infinite stripes and matching with the exterior expansion. We prove the related error estimate.     

Asymptotic expansion for eigenvalue density

The density of eigenvalues of a potential well is calculated in an asymptotic expansion for large geometrical size. Explicit, readily calculable expressions are obtained for volume and surface contributions. The resulting expressions are numerically applied to the case of a spherical Woods-Saxon well.     

A New Inequality for the von Neumann Entropy

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.     

Quantum Marginal Inequalities and the Conjectured Entropic Inequalities

A conjecture – the modified super-additivity inequality of relative entropy – was proposed in Zhang et al. (Phys. Lett. A 377:1794–1796, 2013): There exist three unitary operators \(U_{A}\in \mathrm {U}(\mathcal {H}_{A}), U_{B}\in \mathrm {U}(\mathcal {H}_{B})\) , and \(U_{AB}\in \mathrm {U}(\mathcal {H}_{A}\otimes \mathcal {H}_{B})\) such that $$\mathrm{S}\left(U_{AB}\rho_{AB}U^{\dagger}_{AB}||\sigma_{AB}\right)\geqslant \mathrm{S}\left(U_{A}\rho_{A}U^{\dagger}_{A}||\sigma_{A}\right) + \mathrm{S}\left(U_{B}\rho_{B}U^{\dagger}_{B}||\sigma_{B}\right), $$ where the reference state σ is required to be full-ranked. A numerical study on the conjectured inequality is conducted in this note. The results obtained indicate that the modified super-additivity inequality of relative entropy seems to hold for all qubit pairs.     

Bounds on the unstable eigenvalue for period doubling

Bounds are given for the unstable eigenvalue of the period-doubling operator for unimodal maps of the interval. These bounds hold for all types of behaviour |x| ^r of the interval map near its critical point. They are obtained by finding cones in function space which are invariant under the tangent map to the doubling operator at its fixed point.     

Inequalities for electrostatic energy

The potential energy for a set of a finite number of point charges that are assumed to be aligned is calculated. The lower estimate of the energy is formally obtained by calculating the interactions between neighboring charges, with each interaction considered as attraction. In another (quantum mechanical) approach, the three-dimensional problem of N bodies is studied. It is established that the lower energy level rises in absolute value no faster than N. This result is extended to the case when the particles have different masses.     

Convolution Inequalities for the Boltzmann Collision Operator

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in L ^p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L ² case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some L^s_weak{L^{s}_{weak}} or L ^s . The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.