Semiunitary Transforrnation and Isospectral Hamiltonians in Arbitrary Dimensional Space

By means of the complex Clifford algebra, a new realization of multi-dimensional semiunitary transformation is put forward and then applied to studying the isospectrality of nonrelativistic Hamiltonians of multi-dimensional quantum mechanical systems, in which the generalized Pauli coupling interaction and spin-orbit coupling interaction appear naturally. Moreover, it is shown that the semiunitary operators, together with the Hamiltonian of quantum mechanical system, satisfy the polynomially-deformed angular momentum algebra.

HAMILTONIANS WITH TWO DEGREES OF FREEDOM ADMITTING A SINGLEVALUED GENERAL SOLUTION

Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t） of the general solution is a singlevalued function of the complez time t. In addition to the well known rational potentials V of Hénon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven （three “cubic” plus four “quartic”） rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.

Inverse problem of quadratic time-dependent Hamiltonians

Using an algebraic approach, it is possible to obtain the temporal evolution wave function for a Gaussian wavepacket obeying the quadratic time-dependent Hamiltonian(QTDH). However, in general, most of the practical cases are not exactly solvable, for we need general solutions of the Riccatti equations which are not generally known. We therefore bypass directly solving for the temporal evolution wave function, and study its inverse problem. We start with a particular evolution of the wave-packet, and get the required Hamiltonian by using the inverse method. The inverse approach opens up a new way to find new exact solutions to the QTDH. Some typical examples are studied in detail. For a specific timedependent periodic harmonic oscillator, the Berry phase is obtained exactly.

Impact of symmetrized and Burt—Foreman Hamiltonians on spurious solutions and energy levels of InAs/GaAs quantum dots

We present a systematic investigation of calculating quantum dots （QDs） energy levels using finite element method in the frame of eight-band k · p method. Numerical results including piezoelectricity, electron and hole levels, as well as wave functions are achieved. In the calculation of energy levels, we do observe spurious solutions （SSs） no matter Burt Foreman or symmetrized Hamiltonians are used. Different theories are used to analyse the SSs, we find that the ellipticity theory can give a better explanation for the origin of SSs and symmetrized Hamiltonian is easier to lead to SSs. The energy levels simulated with the two Hamiltonians are compared to each other after eliminating SSs, different Hamiltonians cause a larger difference on electron energy levels than that on hole energy levels and this difference decreases with the increase of QD size.

A method to calculate effective Hamiltonians in quantum information

Effective Hamiltonian method is widely used in quantum information. We introduce a method to calculate effective Hamiltonians and give two examples in quantum information to demonstrate the method. We also give a relation between the effective Hamiltonian in the Shr?dinger picture and the corresponding effective Hamiltonian in the interaction picture.Finally, we present a relation between our effective Hamiltonian method and the James–Jerke method which is currently used by many authors to calculate effective Hamiltonians in quantum information science.

Staggering in Rotor Hamiltonians Without C_(4) Symmetry

After a brief review of the negative results of the microscopic study,of both the Strutinsky shell correction method and Hartree-Fock approximation,for the C_(4) symmetry in nuclei,a proposed rotor Hamiltonian to simulate both the microscopic results and the staggering effect is investigated and the staggering resulting from the rotor Hamiltonians without C_(4) symmetry can be found.

Establishing path integral in the entangled state representation for Hamiltonians in quantum optics

Based on two mutually conjugate entangled state representations, we establish the path integral formalism for some Hamiltonians of quantum optics in entangled state representations. The Wigner operator in the entangled state representation is presented. Its advantages are explained.

Deriving Energy-Gap of Some Nonlinear Hamiltonians by Invariant Eigen-operator Method

By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for someHamiltonians describing nonlinear processes in particle physics.In this way the energy-gap of the Hamiltonians can benaturally obtained.The characteristic polynomial theory has been fully employed in our derivation.

LOOPORBITSFORSINGULARHAMILTONIANSYSTEMS

This paper is motivated by looking for a loop solution of the Hamiltonian systems such that (0.1) q'(t)+V′(q(t))=0 for t∈ with some T>0 and (0.2) 12|q′(t)| 2+V(q(t))=h for t∈ with q(0)=q(T)=x 0 where q∈C 2(, R n 0}), n≥2, x 0∈R n 0} is a fixed point, h∈R is a given number, V∈C 2(R n 0}), R is a potential with a singularity and V′ denotes its gradient. Our main existence results are obtained by a appropriately defined lengthdecreasing (or rather energy decreasing) deformation and a min max procedure which is a combined version of Bahri Rabinowitz and Klingenberg . Our main assumptions are geodesic convex conditions found by the author and the strong force condition of Gordon . As a direct application, for the relativistic gravitational potential V(x)=|x| -1 +|x| -2 or its large scale perturbation, there always exists an almost periodic solution of (0.1)-(0.2) for any h∈R and any x 0∈R n 0} with | x 0 | small enough. This is an interesting phenomenon because we know that there exists no periodic solution of prescribed nonnegative energy for such a Hamiltonian system.

A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in Arbitrary Gravitational Fields and Its Application to Centrally and Axially Symmetric Gravitational Fields

We have proposed previously a method for constructing self-conjugate Hamiltonians Hh in the h-representation with a flat scalar product to describe the dynamics of Dirac particles in arbitrary gravitational fields. In this paper, we prove that, for block-diagonal metrics, the Hamiltonians Hh can be obtained, in particular, using “reduced” parts of Dirac Hamiltonians, i.e. expressions for Dirac Hamiltonians derived using tetrad vectors in the Schwinger gauge without or with a few summands with bispinor connectivities. Based on these results, we propose a modified method for constructing Hamiltonians in the h-representation with a significantly smaller amount of required calculations. Using this method, here we for the first time find self-conjugate Hamiltonians for a number of metrics, including the Kerr metric in the Boyer-Lindquist coordinates, the Eddington-Finkelstein, Finkelstein-Lemaitre, Kruskal, Clifford torus metrics and for non-stationary metrics of open and spatially flat Friedmann models.

Quasi-Exactly Solvable Time-Dependent Hamiltonians

A generalized method which helps to find a time-dependent Schr&#214dinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.

Description of Weakly Bound and Resonant Nuclei with the Gamow Shell Model using Effective and Realistic Hamiltonians

We investigate the theoretical description of nuclei at drip-lines.For this,the Gamow shell model has been developed to study the properties of weakly bound and resonance nuclei.

Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians：The Case of Stellar Halo of Milky Way

Recently,the notion of non-standard Lagrangians was discussed widely in literature in an attempt to explore the inverse variational problem of nonlinear differential equations.Different forms of non-standard Lagrangians were introduced in literature and have revealed nice mathematical and physical properties.One interesting form related to the inverse variational problem is the logarithmic Lagrangian,which has a number of motivating features related to the Li′enard-type and Emden nonlinear differential equations.Such types of Lagrangians lead to nonlinear dynamics based on non-standard Hamiltonians.In this communication,we show that some new dynamical properties are obtained in stellar dynamics if standard Lagrangians are replaced by Logarithmic Lagrangians and their corresponding non-standard Hamiltonians.One interesting consequence concerns the emergence of an extra pressure term,which is related to the gravitational field suggesting that gravitation may act as a pressure in a strong gravitational field.The case of the stellar halo of the Milky Way is considered.

Quantum Fisher Information Gap for Systems with Nonlinear Hamiltonians

Quantum Fisher information(QFI) gap characterizes the stability of QFI to space directions. We study the QFI distributions and QFI gap for quantum states generated from nonlinear Hamiltonians for both spin and bosonic systems. We find that the same spin-squeezing parameter(or principle squeezing parameter) corresponds to two different values QFI gap, and the locations of all extreme points of the QFI are explicitly given.

Equivariant analytical mapping of first principles Hamiltonians to accurate and transferable materials models

We propose a scheme to construct predictive models for Hamiltonian matrices in atomic orbital representation from ab initio data as a function of atomic and bond environments.The scheme goes beyond conventional tight binding descriptions as it represents the ab initio model to full order,rather than in two-centre or three-centre approximations.We achieve this by introducing an extension to the atomic cluster expansion(ACE)descriptor that represents Hamiltonian matrix blocks that transform equivariantly with respect to the full rotation group.The approach produces analytical linear models for the Hamiltonian and overlap matrices.Through an application to aluminium,we demonstrate that it is possible to train models from a handful of structures computed with density functional theory,and apply them to produce accurate predictions for the electronic structure.The model generalises well and is able to predict defects accurately from only bulk training data.

Transferable equivariant graph neural networks for the Hamiltonians of molecules and solids

This work presents an E(3)equivariant graph neural network called HamGNN,which can fit the electronic Hamiltonian matrix of molecules and solids by a complete data-driven method.Unlike invariant models that achieve equivariance approximately through data augmentation,HamGNN employs E(3)equivariant convolutions to construct the Hamiltonian matrix,ensuring strict adherence to all equivariant constraints inherent in the physical system.In contrast to previous models with limited transferability,HamGNN demonstrates exceptional accuracy on various datasets,including QM9 molecular datasets,carbon allotropes,silicon allotropes,SiO_(2) isomers,and BixSey compounds.The trained HamGNN models exhibit accurate predictions of electronic structures for large crystals beyond the training set,including the Moirétwisted bilayer MoS2 and silicon supercells with dislocation defects,showcasing remarkable transferability and generalization capabilities.The HamGNN model,trained on small systems,can serve as an efficient alternative to density functional theory(DFT)for accurately computing the electronic structures of large systems.

Perturbations from a kind of quartic Hamiltonians under general cubic polynomials

In this paper we investigate the perturbations from a kind of quartic Hamiltonians under general cubic polynomials. It is proved that the number of isolated zeros of the related abelian integrals around only one center is not more than 12 except the case of global center. It is also proved that there exists a cubic polynomial such that the disturbed vector field has at least 3 limit cycles while the corresponding vector field without perturbations belongs to the saddle loop case.

The structure of the spherical tensor forces in the USD and GXPF1A shell model Hamiltonians

The realistic shell model Hamiltonians, USD and GXPF1A, have been transformed from the particle-particle （normal） representation to the particle-hole representation （multipole-multipole） by using the known formulation in Ref. [1]. The obtained multipole-multipole terms were compared with the known spherical tensor forces, including the coupled ones. It is the first time the contributions of the coupled tensor forces to the shell model Hamiltonian have been investigated. It has been shown that some coupled-tensor forces, such as [r2Y2 σ] 1, also give important contributions to the shell model Hamiltonian.

Projectability disentanglement for accurate and automated electronic-structure Hamiltonians

Maximally-localized Wannier functions(MLWFs)are broadly used to characterize the electronic structure of materials.Generally,one can construct MLWFs describing isolated bands(e.g.valence bands of insulators)or entangled bands(e.g.valence and conduction bands of insulators,or metals).Obtaining accurate and compact MLWFs often requires chemical intuition and trial and error,a challenging step even for experienced researchers and a roadblock for high-throughput calculations.Here,we present an automated approach,projectability-disentangled Wannier functions(PDWFs),that constructs MLWFs spanning the occupied bands and their complement for the empty states,providing a tight-binding picture of optimized atomic orbitals in crystals.Key to the algorithm is a projectability measure for each Bloch state onto atomic orbitals,determining if that state should be kept identically,discarded,or mixed into the disentanglement.We showcase the accuracy on a test set of 200 materials,and the reliability by constructing 21,737 Wannier Hamiltonians.

Creating locally interacting Hamiltonians in the synthetic frequency dimension for photons

The recent emerging field of synthetic dimension in photonics offers a variety of opportunities for manipulating different internal degrees of freedom of photons such as the spectrum of light.While nonlinear optical effects can be incorporated into these photonic systems with synthetic dimensions,these nonlinear effects typically result in long-range interactions along the frequency axis.Thus,it has been difficult to use the synthetic dimension concept to study a large class of Hamiltonians that involves local interactions.Here we show that a Hamiltonian that is locally interacting along the synthetic dimension can be achieved in a dynamically modulated ring resonator incorporatingχ3nonlinearity,provided that the group velocity dispersion of the waveguide forming the ring is specifically designed.As a demonstration we numerically implement a Bose–Hubbard model and explore photon blockade effect in the synthetic frequency space.Our work opens new possibilities for studying fundamental many-body physics in the synthetic space in photonics,with potential applications in optical quantum communication and quantum computation.