Optimized Lower Bound for Four-Body Hamiltonians

Generalizing a method elaborated for three-body systems, we derive a new lower bound on four-body ground-state energies in terms of two-body binding energies in the unequal-mass case. For simple power-law potentials, this bound is compared to variational calculations and is shown to be very close to the exact result. In particular, it gives the exact answer for harmonic interactions. Received November 6, 1997; accepted in final form February 6, 1998     

Lower Bounds for Nodal Sets of Eigenfunctions

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.     

Lower Bounds for the Robustness of Multilevel Coherence

International Journal of Theoretical Physics - Quantum coherence, coming from quantum superposition, occupies a significant position in the field of physics. We put forward a lower bound of the...     

Lower Bounds for Eigenvalues of the Stokes Operator

In this paper, we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator. We check and prove this condition for four nonconforming methods and one conforming method. Hence they produce eigenvalues which are smaller than their exact counterparts.     

Lower Bounds for Nodal Sets of Dirichlet and Neumann Eigenfunctions

Let ${\phi}$ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set ${\{\phi = 0\}}$ .     

Flow-equations for Hamiltonians

Flow-equations are introduced in order to bring Hamiltonians closer to diagonalization. It is characteristic for these equations that matrix-elements between degenerate or almost degenerate states do not decay or decay very slowly. In order to understand different types of physical systems in this framework it is probably necessary to classify various types of these degeneracies and to investigate the corresponding physical behavior. In general these equations generate many-particle interactions. However, for an n-orbital model the equations for the two-particle interaction are closed in the limit of large n. Solutions of these equations for a one-dimensional model are considered. There appear convergency problems, which are removed, if instead of diagonalization only a block-diagonalization into blocks with the same number of quasiparticles is performed.     

Oracles and Query Lower Bounds in Generalised Probabilistic Theories

We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying four natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle model to be well-defined. The four principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as the marginal of a pure state of a larger system), strong symmetry (existence of a rich set of nontrivial reversible transformations), and informationally consistent composition (roughly: the information capacity of a composite system is the sum of the capacities of its constituent subsystems). Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, because fewer queries provide no information about the solution, then the corresponding “no-information” lower bound in theories lying at the kth level of Sorkin’s hierarchy is \(\lceil {n/k}\rceil \). This lower bound leaves open the possibility that quantum oracles are less powerful than general probabilistic oracles, although it is not known whether the lower bound is achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource that might have power beyond that offered by quantum computation.     

CPT-Frames for Non-Hermitian Hamiltonians

Based on the P T-symmetric quantum theory,the concepts of P T-frame,P T-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed.It is proved that the spectrum and point spectrum of a P T-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken P T-symmetric operator are real.For a linear operator H on Cd,it is proved that H has unbroken P Tsymmetry if and only if it has d diferent eigenvalues and the corresponding eigenstates are eigenstates of P T.Given a C P T-frame on K,a new positive inner product on K is induced and called C P T-inner product.Te relationship between the CP T-adjoint and the Dirac adjoint of a densely defined linear operator is derived,and it is proved that an operator which has a bounded CP T-frame is CP T-Hermitian if and only if it is T-symmetric,in that case,it is similar to a Hermitian operator.The existence of an operator C consisting of a CP T-frame is discussed.These concepts and results will serve a mathematical discussion about P T-symmetric quantum mechanics.     

CPT-Frames for Non-Hermitian Hamiltonians

Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C^d, it is proved that H has unbroken PT- symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.     

Lower Bounds on Negative Energy Densities for the Scalar Field in Flat Spacetime

We obtain a lower bound on the spacetime-weighted average of the energy density for the scalar field in four-dimensional flat spacetime. The bound takes the form of a quantum inequality. The inequality does not rely on the quantum state and its form is only related to the weights, namely the spacetime sampling functions which are assumed to be smooth, positive and compactly supported. It is found that the inequality is just equal to the temporal quantum energy inequality. When the characteristic length of the temporal sampling function tends to zero, the lower bound becomes divergent. This is consistent with the fact that the spatial restriction on negative energy density does not exist in four-dimensional spacetime.     

Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy

This paper studies the properties of the derivatives of differential entropy H(Xt) in Costa’s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for m≥1, (−1)m+1(dm/dtm)H(Xt)≥0, while McKean conjectured a stronger statement, whereby (−1)m+1(dm/dtm)H(Xt)≥(−1)m+1(dm/dtm)H(XGt). Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: C1(m,n):(−1)m+1(dm/dtm)H(Xt)≥0, where n denotes that Xt is a random vector taking values in Rn, and similarly, C2(m,n):(−1)m+1(dm/dtm)H(Xt)≥(−1)m+1(dm/dtm)H(XGt)≥0. In this paper, we prove some new multivariate cases: C1(3,i),i=2,3,4. Motivated by our results, we further propose a weaker version of McKean’s conjecture C3(m,n):(−1)m+1(dm/dtm)H(Xt)≥(−1)m+11n(dm/dtm)H(XGt), which is implied by C2(m,n) and implies C1(m,n). We prove some multivariate cases of this conjecture under the log-concave condition: C3(3,i),i=2,3,4 and C3(4,2). A systematic procedure to prove Cl(m,n) is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.     

Path Integrals for Pseudo-Hermitian Hamiltonians

Inner products in pseudo-Hermitian quantum theories depend on the details of the Hamiltonians themselves, which makes them difficult to calculate. We shall see that, for some questions, the functional integrals for such theories can be calculated without needing to determine the inner product metric. The reason is that their derivation is based on the Heisenberg equations of motion and the canonical commutation relations, which are unchanged. In particular, this can greatly simplify the derivation of Hermitian theories that are equivalent to these pseudo-Hermitian systems.     

Improved Lower Bounds for the Critical Probability of Oriented Bond Percolation in Two Dimensions

We present a coupled decreasing sequence of random walks on Z that dominate the edge process of oriented bond percolation in two dimensions. Using the concept of random walk in a strip, we describe an algorithm that generates an increasing sequence of lower bounds that converges to the critical probability of oriented percolation p_c. From the 7th term on, these lower bounds improve upon 0.6298, the best rigorous lower bound at present, establishing 0.63328 as a rigorous lower bound for p_c. Finally, a Monte Carlo simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for p_c. Mathematics Subject Classification (1991): 60K35 Supported by CNPq (grant N.301637/91-1). Supported by a grant from CNPq.     

Boltzmann Model for Viscoelastic Particles: Asymptotic Behavior,Pointwise Lower Bounds and Regularity

We investigate the long-time behavior of a system of viscoelastic particles modeled with the homogeneous Boltzmann equation. We prove the existence of a universal Maxwellian intermediate asymptotic state with explicit rate of convergence towards it. Exponential lower pointwise bounds and propagation of regularity are also studied. These results can be seen as a generalization of several classical results holding for the pseudo-Maxwellian and constant normal restitution models.     

Exact solutions for nonlinear Hamiltonians

We find the eigenvalues and eigenvectors of two nonlinear Hamiltonians describing the interaction between a two-level system and a quantized linear harmonic oscillator. In the first case we obtain exact isolated solutions for the Hamiltonian used as a model of an ion in a harmonic trap and interacting with a laser field, not restricted to the Lamb-Dicke limit. After projecting these eigenstates onto one of the levels of the two-level system the oscillator state is described by a finite superposition of Fock states. In the second case we consider a Hamiltonian, with a squeeze operator in the interaction part. We give perturbation results in the weak-coupling limit and results obtained by numerical diagonalization for the strong coupling limit. Non-classical results are pointed out also in this case.     

Canonical forms for quadratic Hamiltonians

Our analysis of the applicable representations of the group of Bogoliubov transformations shows that the diagonalization of a quadratic fermion Hamiltonian with arbitrary complex coefficients is equivalent to the reduction of a skew symmetric matrix to secondary diagonal form by an orthogonal transformation, which we construct explicitly.Similarly, the diagonalization of a positive definite boson Hamiltonian with complex coefficients is equivalent to Whittaker's diagonalization of a symmetric matrix by a symplectic transformatio. Both results are shown to follow from a general spectral theorem for indefinite inner product space, a recent extension of which allows us to block diagonalize a positive definite quadratic boson Hamiltonian with complex coefficients and infinite degrees of freedom and thereby provide a counterpart to Araki's result for Fermi fields.     

Polynomial supersymmetry for matrix Hamiltonians

We study intertwining relations for matrix non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any intertwining operator of minimal order there is operator that intertwines the same Hamiltonians in the opposite direction and such that the products of these operators are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and reducibility of a matrix intertwining operator are considered and the criteria of minimizability and reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.     

Positivity Representations for Non-Hermitian Hamiltonians

We show how 1-dimensional, non-Hermitian, Sturm-Liouville systems, with rational fraction potentials, define a Moment Problem positivity representation, enabling the generation of (numerical and algebraic) converging lower and upper bounds to the (real and imaginary parts of) complex eigenenergies.