Spectral shift functions (SSFs) provide a powerful framework for understanding how the spectrum of a self-adjoint operator changes under perturbation, and they play a central role in trace formulas that generalize the classical results of Krein and Koplienko. While higher-order SSFs have been extensively developed in abstract settings—particularly under Schatten class assumptions or within noncommutative frameworks with τ-compact resolvents—their explicit computation remains challenging, especially for differential operators arising in quantum mechanics. In this paper, we define and compute the SSF of order k for a one-dimensional Schrödinger operator perturbed by a constant potential, and rigorously verify the associated trace formula. Our approach, grounded in classical Hilbert–Schmidt theory and Fourier analysis, bypasses abstract machinery while capturing physically meaningful scenarios. These results also serve as a foundation for future work on more singular perturbations, such as delta and square-well potentials.

Self-Interacting Quantum Particles and the Dirac Delta Potential

Abstract We present a unified analytical study of the transmission phase time for Dirac particles scattered by delta-function potentials in one and two dimensions. For the one-dimensional case, we obtain a closed-form expression for the transmission relativistic phase time and show that it coincides with both the reflection phase time and the self-interference delay, in agreement with general predictions for symmetric, zero-width scatterers. The resulting delay alternates periodically between positive and negative values as the coupling strength increases, and converges to the known non-relativistic expression in the appropriate limit. This delay-advance pattern can be interpreted through the spectral structure of the Dirac Hamiltonian, where electron and positron bound-state sectors correlate with time advances and delays, respectively, reflecting the underlying nature of the delta interaction. In the two-dimensional case, we derive an exact expression for the transmission phase time in graphene and examine its angular and coupling dependence. The delay exhibits a periodic, butterfly-like interference pattern governed by resonant transmission and Klein tunnelling, vanishing at normal incidence and at integer values of the reduced coupling strength. These features reveal how angular interference and relativistic dynamics shape temporal signatures of quantum transport in Dirac materials. The closed-form analytical expressions for the relativistic phase time in one dimension and in two dimensions (graphene), which constitute the main results of this work. Our results provide analytic insight into phase-time phenomena in relativistic systems and provide compact benchmarks for time-resolved quantum transport in narrow-gap and Dirac-like media.

Abstract We theoretically study the conductance spectrum of metal/semiconductor/metal junction incorporating direct Rashba–Dresselhaus spin–orbit interaction (RDSOI). The system is modeled using Dirac delta-function potentials to represent interface mismatches. Both single- and double-junction configurations are considered. In the single-junction system, the conductance exhibits a sharp increase near the flat-band energy of the RDSOI region and gradually decreases with increasing bias voltage. This behavior is strongly dependent on the interface transparency, quantified by the barrier strength. While varying the relative strengths of Rashba and Dresselhaus interaction does not qualitatively alter the conductance spectrum, an overall enhancement in spin–orbit interaction reduces the threshold bias voltage. For the double-junction system, quantum interference effects within the RDSOI region give rise to bias-dependent conductance oscillations. The amplitude and frequency of these oscillations are governed by the length of the RDSOI segment and the interface potential. Additionally, increasing the barrier height suppresses conductance and shifts the resonance peaks, reflecting modified tunneling conditions.

The single and double Dirac delta potentials under the assumption of minimal length

Abstract Put options are known to be priced unusually high in the market, which we refer to as the overpriced put puzzle . This study proposes a quantum model (QM) that can explain such high put option prices as fair prices. Starting from a stochastic differential equation of stock returns, we convert the Fokker–Planck equation into the Schrödinger equation. To model the market force that always draws excess returns back to equilibrium, we specify a diffusion process corresponding to a QM with a delta potential. The results demonstrate that stock returns follow a Laplace distribution and exhibit power law in the tail. We then construct a closed-form solution for European put option pricing, determining that our model better explains the returns of the S&P 500 index and its corresponding put option prices than do geometric Brownian motion-based models. This study has significant implications for investors and risk managers, presenting a model that can potentially improve derivative pricing. Future studies can generalize the model assumptions by introducing asymmetric potential drawing back excess returns to equilibrium.

In this work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made precise by a renormalization scheme) supported at a point in two- and three-dimensional compact manifolds or Euclidean spaces. The formulation can be easily extended to an N center case and the case where delta interaction is supported on curves in the plane or space. We finally give an interesting application for the sudden perturbation of the support of the delta potential.

Optimal error estimate of a conservative, efficient and accurate finite difference scheme for the nonlinear Schrödinger equation with Dirac delta potentials

The exact analytical solution of the one-dimensional stationary Schrödinger equation with a spatially distributed hyperbolic potential profile ~1/x in the presence of a Dirac delta function (point) potential is found and analyzed. Localization features are described analytically in dependence on the parameters of a hyperbolic potential profile and point potential. It is found that the localization energy of the ground state monotonically decreases with an increase in defect intensity. The localized states exist near the both attractive and repulsive defects. Excited states are characterized by the formation of oscillations in the region of action of the hyperbolic field. The height of the maximum of the wave function decreases with decreasing intensity of the repulsive defect and with increasing absolute value of the defect intensity in the case of attractive defect. Localization length in the hyperbolic well is greater than in a halfspace with a constant potential.

A novel transfer matrix framework for multiple Dirac delta potentials

This paper delves into entanglement-related features of one-dimensional and three-dimensional systems comprising two particles interacting through an attractive potential, such as the delta, harmonic, and Coulomb ones. As a quantitative indicator of the amount of entanglement between the particles, we employ the linear entropy of the system’s one-particle marginal density matrices. Except in some particular instances involving the harmonic potential, this quantity is not analytically tractable and requires numerical evaluation. Our aim is to elucidate some aspects of entanglement in hydrogenic systems. Hydrogenic systems, consisting of two particles interacting through the Coulomb potential, are of clear importance in physics and chemistry, but their entanglement properties have started to be explored only recently. To better understand entanglement in those systems, we first analyze one-dimensional systems interacting via Dirac delta and harmonic potentials. Insights gained from these one-dimensional cases provide valuable guidelines for studying entanglement in hydrogenic systems. We numerically investigate, for the interaction potentials already mentioned, and for different types of confinement for the center of mass, how the system’s entanglement varies with the parameters that determine the size and geometry of the system’s quantum state. We find that entanglement depends on a dimensionless quantity given by the quotient of two parameters characterizing the length scales associated with the interaction potential and the center of mass confinement. Entanglement approaches its maximum when the above-mentioned dimensionless quotient tends to its extreme values and adopts its minimum at an intermediate value of the dimensionless quotient. We find that the same general qualitative features of entanglement behavior are observed for different attractive interactions.

The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.

Understanding the effect that impurities may have on the soliton propagation process, particularly during the interaction process involving the Nonlinear Schrödinger Equation (NLSE), has become a major research focus in recent years. This paper studied the phenomenon of soliton scattering when it interacts with a localized impurity of the Delta potential under the discrete case of NLSE. Using an analytical approach, i.e., the variational approximation (VA) method, the equations of soliton parameters for the width, center-of-mass position, and linear and quadratic phase-front corrections are derived in order to describe the soliton evolutions throughout the scattering process. The VA method results were validated by the direct numerical simulation of the discrete NLSE, provided that the soliton is initially set at a distance from the Delta potential. When the nonlinearity was taken to be of the cubic and quintic types, it was shown that the soliton of the Discrete Cubic-Quintic NLSE could be either reflected or transmitted by the Delta potential with different potential strengths and a constant soliton’s initial velocity. The results suggested that the VA method is an effective and useful approach to investigate the scattering process of discrete NLSE in the presence of impurities.

We prove the existence of multi-soliton solutions for the nonlinear Schrödinger equation with repulsive Dirac delta potential and L2-supercritical focusing nonlinear term. Our main contribution is to treat the unmoving part of the multi-solitons, which is the ground state of the equation. The linearized operator around it has two unstable eigenvalues. This is the main difference from NLS without potential, whose existence of multi-solitons is investigated by Côte et al. (Rev Mat Iberoam 27:273–302, 2011).

Transport Properties in ZnO/ZnCdO Heterostructure with Spin–Orbit Interaction: Effect of In-Plane Magnetic Field & Delta Potential

Long-time asymptotics of the damped nonlinear Klein–Gordon equation with a delta potential

Stability is an essential problem in theoretical and experimental studies of solitons in nonlinear media with fractional diffraction, which is represented by the Riesz derivative with Lévy index (LI) α, taking values α<2. Fractional solitons are unstable at α≤1 or α≤2 in uniform one-dimensional media with the cubic or quintic self-focusing, respectively. We demonstrate that, in these cases, the solitons may be effectively stabilized by pinning to a delta-functional trapping potential (attractive defect), which is a relevant setting in optical waveguides with the effective fractional diffraction. Using the respective fractional nonlinear Schrödinger equation with the delta-functional potential term, we find that, in the case of the cubic self-focusing, the fractional solitons are fully stabilized by the pinning to the defect for α=1 and partly stabilized for α<1. In the case of the quintic self-focusing, the full and partial stabilization are found for α=2 and α<2, respectively. In both cases, the instability boundary is exactly predicted by the Vakhitov-Kolokolov criterion. Unstable solitons spontaneously transform into oscillating breathers. A variational approximation (VA) is elaborated parallel to the numerical analysis, with a conclusion that the VA produces accurate results for lower LI values, i.e., stronger fractionality. In the cubic medium, collisions of traveling stable solitons with repulsive and attractive defects are addressed too, demonstrating outcomes in the form of rebound, splitting, and passage.

In this work, we have investigated the Gaussian and Airy wave packets colliding on the Dirac delta potentials. We have used the time-dependent space-fractional Schrodinger equation and solved it by using the slit step finite difference method. Here, we have presented some exclusive exotic propagation properties of Dirac delta potential that cannot be observed for other very similar potentials such as one-dimensional coulomb potential. We can now rectify any number of colliding wave packets provided that the correct parameters adjustment is possible. We have also addressed the wave packet speed control, wave packet splitting, wave packet rectification through collision process, wave packet dispersion, etc. The effects of different parameters such as the nonlinearity, fractitionality, dispersion, etc. on the wave packet-delta potential collision are also considered.

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 eigenenergies as a function of particle quasimomentum, which form band structure, as in standard Kronig-Penney model. We test our model by comparing it to the numerical calculations for an atom scattering on an infinite chain of ions in quasi-1D geometry. The agreement is fairly good and can be further improved by introducing energy-dependent scattering length in the regularized delta potential. The energy spectrum exhibits the presence of multiple overlapping bands resulting from excitations in the transverse direction. At large lattice constants, our model reduces to standard Kronig-Penney result with 1D coupling constant for quasi-1D scattering, exhibiting confinement-induced resonances. In the opposite limit, when lattice constant becomes comparable to harmonic oscillator length of the transverse potential, we calculate the correction to the quasi-1D coupling constant due to the quantum interference between scatterers. Finally, we calculate the effective mass for the lowest band and show that it becomes negative for large and positive scattering lengths.

Abstract We study the spectrum, eigenstates and transport properties of a simple P T -symmetric model consisting in a finite, complex, square well potential with a delta potential at the origin. We show that as the strength of the delta potential increases, the system exhibits exceptional points accompanied by an accumulation of density associated with the break in the P T -symmetry. We also obtain the density and energy density fluxes and analyze their transport properties. We find that in the P T -symmetric phase transport is efficient, in the sense that all the density that flows into the system at the source, flows out at the sink, which is sufficient to derive a generalized unitary relation for the transmission and reflection coefficients.