Variational Bound States of Screened Potentials

Anumber of years ago, a calculational scheme was introduced by Stubbins [Phys. Rev. A48, 220 (1993)] to compute the energies of both the Hulthén and Yukawa potentials. The method introduces a particular ansatz for solving the Schrödinger equation with screened Coulomb type potentials. In this work, we wish to review the method of Stubbins and to show that it is, in fact, equivalent and a subset of a more systematic (and hence more useful) variational scheme [Zhou et al. Phys. Rev. A51, 3337 (1995)]. This variational approach involves the construction of a basis by taking derivatives of the variational parameters of the system. The eigenvalues of the Hamiltonian matrix are then minimized with respect to these parameters yielding a “best guess” upper bound on the energies.     

Discussion on Weakly Bound States of Three-Body Systems

The equivalent two-body method for a three-body system has been generalized to an arbitrary three-body system with short-range two-body interactions. An analytical expression for the long-range effective potential is obtained for the Gaussian potential, the Yukawa potential and the exponential potential. The asymptotic behavior of the effective potential at very large distance is found to be universal and an explanation on the significance of universality is given. The weakly bound excited state for the system is first obtained although there is no bound state for two-body subsystems.     

Fugacity and Density Expansion for Systems with Bound States

The fugacity expansion in different approximations for low densities and for systems with bound and scattering states is discussed. The meaning of a mixed expansion and the possibility to introduce a chemical picture for a system in which bound states dominate is proved. The possibility of a thermodynamical instability is shown.     

Multiple Bound States of Nonlinear Schrödinger Systems

This paper is concerned with existence of bound states for Schrödinger systems which have appeared as several models from mathematical physics. We establish multiplicity results of bound states for both small and large interactions. This is done by different approaches depending upon the sizes of the interaction parameters in the systems. For small interactions we give a new approach to deal with multiple bound states. The novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows. For large interactions we use a minimax procedure to distinguish solutions by analyzing their Morse indices.     

Bound States in Coulomb Systems ‐ Old Problems and New Solutions

We analyze the quantum statistical treatment of bound states in Hydrogen considered as a system of electrons and protons. Within this physical picture we calculate analytically isotherms of pressure for Hydrogen in a broad density region and compare to some results from the chemical picture. Our study is restricted to the range of intermediate temperatures 10⁴K < T < 10⁵K and not too high densities n < 10²⁴ protons per cm³, the formation of molecules is neglected. First we resume in detail the two transitions along isotherms: (i) formation of bound states occurring by increasing the density from low to moderate values, (ii) the destruction of bound states in the high density region, modelled here by Pauli‐Fock effects. Avoiding chemical models we will show, why bound states according to a discrete part of the spectra occur only in a valley in the T‐p plane. First we study virial expansions in the canonical ensemble and then in the grand canonical ensemble. We show that in fugacity representations the population of bound states saturates at higher density and that a combination of both representations provides quickly converging equations of state. In the case of degenerate systems we calculated first the density‐dependent energy levels, and find the pressure in Hartree‐Fock‐Wigner approximation showing the prominent role of Pauli blocking and Fock effects in the selfenergy (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of Three- and Four-Charge Systems

 A brief review is presented of the stability domain of three- and four-charge ground states when the constituent masses vary. Rigorous results are presented, based on the scaling behaviour and the convexity properties deduced from the variational principle. They are supplemented by accurate numerical computations. Received October 25, 2001; accepted for publication November 15, 2001     

Variational Quantum Algorithms for the Steady States of Open Quantum Systems

The solutions of the problems related to open quantum systems have attracted considerable interest.We propose a variational quantum algorithm to find the steady state of open quantum systems.In this algorithm,we employ parameterized quantum circuits to prepare the purification of the steady state and define the cost function based on the Lindblad master equation,which can be efficiently evaluated with quantum circuits.We then optimize the parameters of the quantum circuit to find the steady state.Numerical simulations are performed on the one-dimensional transverse field Ising model with dissipative channels.The result shows that the fidelity between the optimal mixed state and the true steady state is over 99%.This algorithm is derived from the natural idea of expressing mixed states with purification and it provides a reference for the study of open quantum systems.     

Four-Nucleon States in the Continuum: A Means to Probe the Nucleon-Nucleon Interaction

 Four-nucleon states in the continuum are studied through exact microscopic calculations based on the solution of the AGS equations for four nonrelativistic quantum particles. Our studies include calculations of cross sections and analyzing powers for all two-body reactions of interest, but here we only show results for n ³He → n ³He. The NN interactions we use are Bonn-CD, Nijmegen II, and Bonn-B. Compared to existing quality data, one finds large discrepancies and some sensitivity to the choice of NN force model. The calculated n + ³He elastic phase shifts show a very strong inelastic resonance at about 0.3 MeV which is not supported by the total cross-section data. This result is due to the existence of a ³ P ₀ (0⁻) resonance in isospin I = 0 at this energy and the undesirable coincidence of n + ³He and p + ³H thresholds in our calculation due to lack of Coulomb repulsion between protons. This interpretation is supported by R-matrix analyses of the data on the basis of coincident thresholds. Calculated 0⁺ and 0⁻ states are compared with modified R-matrix analyses. Received October 30, 2001; accepted for publication November 7, 2001     

Universality of Three-Body Systems in 2D: Parametrization of the Bound States Energies

Universal properties of mass-imbalanced three-body systems in 2D are studied using zero-range interactions in momentum space. The dependence of the three-particle binding energy on the parameters (masses and two-body energies) is highly non-trivial even in the simplest case of two identical particles and a distinct one. This dependence is parametrized for ground and excited states in terms of supercircles functions in the most general case of three distinguishable particles.     

Bound States in Yukawa Theory

A generalization of the Gell-Mann–Low Theorem is applied to lowest nontrivial order to bound state calculations in Yukawa theory. We present the solution of the corresponding effective Schrödinger equation for two-fermion bound states with the exchange of a massless boson. The complete low-lying bound state spectrum is obtained for fine structure constants below one and different ratios of the constituent masses. The consistency of the nonrelativistic and one-body limits is explicitly verified.     

The Expansion Method of Mixed Basis Vectors of Lower-energy States and Harmonic Oscillators in Calculation of Bound States of Few-Body Systems

In this paper,the harmonic oscillator approach to the bound states of few-body ststems is developed and the lower-energy states are introduced as basis vectors and mixed with a part of harmonic oscillator vectors to calculate the binding energy.The lower energy levels of 3-α system and ⁹Be are presented and compared with experiments or other calculations.The results are satisfactory.     

Weakly Bound States in Light Hypernuclei

From the viewpoint of critical stability, we discuss the three- and four-body structure of ⁶He, ⁷He, ⁴He, and ⁴H. With the + + N three-body model, ⁶He is found to have a three-layer structure of the matter distribution: core, a skin and neutron halo. Also the level structure of ⁷He with the three-body model of ⁵He + n + n is predicted. This stimulates a new study of neutron-rich and proton-rich hypernuclei. By performing a four-body calculation with both NNN and NNN channels and with both NN and NNN channels, we show that the N-N and -N couplings are very important in critical stability of few-body hypernuclear systems.     

Variational Functions in Open Systems

This paper looks for an entropy-like quantity having a monotonic time development. In the case of spontaneous emission, the final state usually consists of a single ground state assigning zero to the ordinary expressions for entropy. Thus entropy ceases to be a monotonic measure of the direction of time. The point is illustrated by a simple test case consisting of three levels coupled by spontaneous emission. It is shown how this case allows the definition of a monotonic function. Using the theory of non-Hermitian operators, the paper shows how such a function may be constructed in the general case, and it explores the main consequences of the expressions suggested. The generalization of the entropy concept is found to relate to time-reversal properties of the dynamics. The paper concludes by discussing open questions and possible further explorations.     

Bound States in Curved Quantum Layers

We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in ℝ³. We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hard-wall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of geometrically induced bound states. Received: 26 February 2001 / Accepted: 21 May 2001     

Schwinger Functions and Light-Quark Bound States

We examine the applicability and viability of methods to obtain knowledge about bound states from information provided solely in Euclidean space. Rudimentary methods can be adequate if one only requires information about the ground and first excited state and assumptions made about analytic properties are valid. However, to obtain information from Schwinger functions about higher mass states, something more sophisticated is necessary. A method based on the correlator matrix can be dependable when operators are carefully tuned and errors are small. This method is nevertheless not competitive when an unambiguous analytic continuation of even a single Schwinger function to complex momenta is available.     

D-Particle Bound States and Generalized Instantons

We compute the principal contribution to the index in the supersymmetric quantum mechanical systems which are obtained by reduction to 0 + 1 dimensions of , D= 4,6,10 super-Yang–Mills theories with gauge group SU(N). The results are: for D=4,6, for D=10. We also discuss the D=3 case. Received: 12 March 1999/ Accepted: 16 July 1999     

Transfer Matrix Spectrum and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We obtain new properties of general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high-temperature region (1). Each model is characterized by a single-site a priori spin distribution, taken to be even. We state our results in terms of the parameter =s ⁴–3s ²², where s ^k denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory which is known to contain particles. We show that for >0, small, there exists a bound state with mass below the two-particle threshold. The existence of the bound state has implications for the decay of correlations, i.e., the 4-point functions decay at a slower rate than twice that of the 2-point function. These results are obtained using a lattice version of the Bethe–Salpeter equation. The existence results generalize to N-component models with rotationally invariant a priori spin distributions.