O. V. Man'ko
There are discussed the exact solution of the time--dependent Schrödinger equation for a damped quantum oscillator subject to a periodical frequency delta--kicks describing squeezed states which are expressed in terms of Chebyshev polynomials. The cases of strong and weak damping are investigated in the frame of Caldirola--Kanai model.
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Photonic analogues of the relativistic Kronig-Penney model and of relativistic surface Tamm states are proposed for light propagation in fibre Bragg gratings (FBGs) with phase defects. A periodic sequence of phase slips in the FBG realizes the relativistic Kronig-Penney model, the band structure of which being mapped into the spectral response of the FBG. For the semi-infinite FBG Tamm surface states can appear and can be visualized as narrow resonance peaks in the transmission spectrum of the grating.
We study the influence of the different choice of unit cells on the Bloch solutions of Schrçdinger equation for one-dimensional periodic Kronig- Penney models with rectangular potential barriers or potential wells and partially constant effectiv mass. We obtain generalized Kronig-Penney relations for bulk states and exact expressions for the corresponding periodic parts of the Bloch wave functions for the two possible choices of the matching conditions in an unit cell. We show also that our analitic expressions reduce to the well known expressions for the Kronig-Penney relations and Bloch wave…
Exact solutions of the time-dependent Schrodinger equation for a quantum oscillator subject to periodical frequency delta-kicks are obtained. We show that the oscillator occurs in the squeezed state and calculate the corresponding squeezing coefficients and the energy increase rate in terms of Chebyshev polynomials.
Nonlinear Kronig-Penney model has been frequently employed to study transmission problem of electron wave in a nonlinear electrified chain or in a doped semiconductor superlattice. Here from an integral equation we derive a novel exact solution of the problem, which contains a simple nonlinear map connecting transmission coefficient with system parameters. Consequently, we suggest a scheme for manipulating electronic distribution and transmission by adjusting the system parameters. A new effect of quantum coherence is evidenced in the strict expression of transmission coefficient by which for …
The standard Kronig-Penney model with periodic $δ$ potentials is extended to the cases with generalized contact interactions. The eigen equation which determines the dispersion relation for one-dimensional periodic array of the generalized contact interactions is deduced with the transfer matrix formalism. Numerical results are presented which reveal unexpected band spectra with broader band gap in higher energy region for generic model with generalized contact interaction.
We introduce a double-stranded Kronig-Penney model and analyze its transport properties. The asymmetric fluxes between two strands with suddenly alternating localization patterns are found as the energy is varied. The zero-size limit of the internal lines connecting two strands is examined using quantum graph vertices with four edges. We also consider a two-dimensional Kronig-Penney lattice with two types of alternating layers with $δ$ and $δ'$ connections, and show that the existence of energy bands in which the quantum flux can flow only in selected directions.
Analytical Solution to the Kronig-Penney Model with Harmonic Oscillator Wells: Insights to Tight-Binding
本文用截断谐振子势替代传统方井势，推导了Kronig-Penney模型的能谱和波函数，揭示了其与传统模型的异同，并推导出紧束缚中隧道幅值的表达式。
The Kronig-Penney model describes what happens to electron states when a confining potential is repeated indefinitely. This model uses a square well potential; the energies and eigenstates can be obtained analytically for a the single well, and then Bloch's Theorem allows one to extend these solutions to the periodically repeating square well potential. In this work we describe how to obtain simple numerical solutions for the eigenvalues and eigenstates for any confining potential within a unit cell, and then extend this procedure, with virtually no extra effort, to the case of periodically re…
We use an exact solution to the fundamental finite Kronig-Penney model with arbitrary positions and strengths of scattering sites to show that this iconic model can possess topologically non-trivial properties. By using free parameters of the system as extra dimensions we demonstrate the appearance of topologically protected edge states as well as the emergence of a Hofstadter butterfly-like quasimomentum spectrum, even in the case of small numbers of scattering sites. We investigate the behaviour of the system in the weak and strong scattering regimes and observe drastically different shapes …
We generalize the textbook Kronig-Penney model to realistic conditions for a quantum-particle moving in the quasi-one-dimensional (quasi-1D) waveguide, where motion in the transverse direction is confined by a harmonic trapping potential. Along the waveguide, the particle scatters on an infinite array of regularized delta potentials. Our starting point is the Lippmann-Schwinger equation, which for quasi-1D geometry can be solved exactly, based on the analytical formula for the quasi-1D Green's function. We study the properties of eigen-energies as a function of particle quasi-momentum, which f…
Chiral and pair superfluidity in triangular ladder produced by state-dependent Kronig-Penney lattice
该研究通过spin依赖性Kronig-Penney晶格实现三角形阶梯，探索了几何摩擦和可控配对跃迁，发现配对超流体和chiral超流体，并确定了XXZ自旋模型的相变点。
We study the properties of a quantum particle interacting with a one dimensional structure of equidistant scattering centres. We derive an analytical expression for the dispersion relation and for the Bloch functions in the presence of both even and odd scattering waves within the pseudopotential approximation. This generalises the well-known solid-state physics text-book result known as the Kronig-Penney model. Our generalised model can be used to describe systems such as degenerate Fermi gases interacting with ions or with another neutral atomic species confined in an optical lattice, thus e…
The Kronig-Penney model, an exactly solvable one-dimensional model of crystal in solid physics, shows how the allowed and forbidden bands are formed in solids. In this paper, we study this model in the presence of both strong spin-orbit coupling and the Zeeman field. We analytically obtain four transcendental equations that represent an implicit relation between the energy and the Bloch wavevector. Solving these four transcendental equations, we obtain the spin-orbital bands exactly. In addition to the usual band gap opened at the boundary of the Brillouin zone, a much larger spin-orbital band…
The main purpose of the present paper is to introduce a scattering approach to the study of the Kronig-Penney model in a quadratic channel with $δ$ interactions, which was discussed in full generality in the first paper of the present series. In particular, a secular equation whose zeros determine the spectrum will be written in terms of the scattering matrix from a single $δ$. The advantages of this approach will be demonstrated in addressing the domain with total energy $E\in [0,\frac{1}{2})$, namely, the energy interval where, for under critical interaction strength, a discrete spectrum is …
The delta function potential is a simple model of zero-range contact interaction in one dimension. The Kronig-Penney model is a one-dimensional periodic array of delta functions that models the energy bands in a crystal. Here we investigate contact interactions that generalize the delta function potential and corresponding generalizations of the Kronig-Penney model within conventional and parity-time symmetric quantum mechanics (PTQM). In conventional quantum mechanics we determine the most general contact interaction compatible with self-adjointness; in PTQM we consider interactions that are …
Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schrödinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining t…
We investigate the transmission properties of a quantum one-dimensional periodic system of fixed length $L$, with $N$ barriers of constant height $V$ and width $λ$, and $N$ wells of width $δ$. In particular, we study the behaviour of the transmission coefficient in the limit $N\to \infty$, with $L$ fixed. This is achieved by letting $δ$ and $λ$ both scale as $1/N$, in such a way that their ratio $γ= λ/δ$ is a fixed parameter characterizing the model. In this continuum limit the multi-barrier system behaves as it were constituted by a unique barrier of constant height $E_o=(γV)/(1+γ)$. The anal…
Zak phase and bulk-boundary correspondence in a generalized Dirac-Kronig-Penney model
本文研究了广义Dirac-克诺格-彭内模型的拓扑性质，发现Zak相在不同对称类中表现出不同的量化特性，并探讨了体-边对应关系与截断参数的关系。
One-dimension Periodic Potentials in Schrödinger Equation Solved by the Finite Difference Method
本研究采用带周期边界条件的有限差分法求解一维Kronig-Penney势，提升了本征值精度，分析了势参数对本征值和波函数的影响，并验证了该方法对狄拉克梳状势的计算与解析解完全吻合。
This paper is dedicated to an application of a quantum wave impedance approach for a study of infinite and semi-infinite periodic systems. Both a Dirac comb and a $δ-δ'$ comb as well as a Kronig-Penney model are considered. It was shown how to reformulate the problem of an investigation of mentioned systems in terms of a quantum wave impedance and it was demonstrated how much a quantum wave impedance approach simplifies studying these systems compared to other methods. The illustation of such a simplification was provided by application of classical approach, transfer matrix technique and a qu…
Quantum transmission in 1D disordered stealthy hyperuniform Kronig--Penney-like models
本文提出一维无序隐身超均匀点散射体量子传输的微扰理论，证明隐身条件与超均匀约束导致低阶项抵消及四阶项抑制，揭示无序SHU系统中波传播的普适规律。
We discuss generation of subwavelength optical barriers on the scale of tens of nanometers, as conservative optical potentials for cold atoms. These arise from non-adiabatic corrections to Born-Oppenheimer potentials from dressed `dark states' in atomic $Λ$-configurations. We illustrate the concepts with a double layer potential for atoms obtained from inserting an optical subwavelength barrier into a well generated by an off-resonant optical lattice, and discuss bound states of pairs of atoms interacting via magnetic dipolar interactions. The subwavelength optical barriers represent an optica…
The known correspondence between the Kronig-Penney model and certain Jacobi matrices is extended to a wide class of Schroedinger operators on graphs. Examples include rectangular lattices with and without a magnetic field, or comb-shaped graphs leading to a Maryland-type model.
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a dam…
Floquet states have been used to describe the impact of periodic driving on lattice systems, either using a tight-binding model, or by using a continuum model where a Kronig-Penney-like description has been used to model spatially periodic systems in one dimension. A number of these studies have focused on finite systems, and results from these studies are distinct from those of infinite lattice systems as a consequence of boundary effects. In the case of a finite system, there remains a discrepancy in the results between tight-binding descriptions and continuous lattice models. Periodic drivi…
Discrete-time quantum walks with energy-dependent coins
该研究将SQW框架扩展到能量依赖点相互作用晶格，构建了对应Kronig-Penney型点相互作用阵列的DTQW模型，分析其谱与动力学，揭示四类点相互作用的不同透射与纠缠行为。
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