arXiv subjects

Mohammad Hasan

Publications and source records attributed to Mohammad Hasan.

At least 19 recordsLinked to original sources

arXiv2026

L\'{e}vy-index control of spectral singularities and coherent perfect absorption in non-Hermitian space-fractional quantum mechanics

We investigate the scattering features of a non-Hermitian rectangular potential within the framework of space-fractional quantum mechanics. Using the Riesz fractional derivative, we analytically derive locus equations for spectral singularities (SSs) and their time-reversed counterparts, coherent perfect absorption (CPA), in a dimensionless complex-potential parameter space. This geometric locus formulation provides a transparent representation of the SS and CPA conditions and enables direct visualization of how fractional quantum dynamics modifies non-Hermitian scattering. We show that reducing the L\'{e}vy index $\alpha$, which enhances nonlocal transport associated with L\'{e}vy-flight dynamics, systematically lowers the gain-loss strength required for the emergence of SSs and CPAs, while increasing the mode index further suppresses this threshold. In addition, for fixed potential parameters, we demonstrate that decreasing $\alpha$ induces a blue shift of the SS energy, in direct agreement with earlier studies. From this perspective, the L\'{e}vy index $\alpha$ emerges as a tunable control knob for SS-CPA settings in fractional non-Hermitian quantum systems. Beyond its quantum-mechanical setting, this study may find applications in fractional waveguides and metamaterials governed by fractional wave equations. This work also bridges the gap between non-Hermitian quantum mechanics and space-fractional quantum mechanics.

quant-ph↗

arXiv2024

Polyadic Cantor potential of minimum lacunarity: Special case of super periodic generalized unified Cantor potential

To bridge the fractal and non-fractal potentials we introduce the concept of generalized unified Cantor potential (GUCP) with the key parameter $N$ which represents the potential count at the stage $S=1$. This system is characterized by total span $L$, stages $S$, scaling parameter $\rho$ and two real numbers $\mu$ and $\nu$. Notably, the polyadic Cantor potential (PCP) system with minimal lacunarity is a specific instance within the GUCP paradigm. Employing the super periodic potential (SPP) formalism, we formulated a closed-form expression for transmission probability $T_{S}(k, N)$ using the $q$-Pochhammer symbol and investigated the features of non-relativistic quantum tunneling through this potential configuration. We show that GUCP system exhibits sharp transmission resonances, differing from traditional quantum systems. Our analysis reveals saturation in the transmission profile with evolving stages $S$ and establishes a significant scaling relationship between reflection probability and wave vector $k$ through analytical derivations.

quant-ph↗

arXiv2024

Detecting Anomalies in Blockchain Transactions using Machine Learning Classifiers and Explainability Analysis

As the use of Blockchain for digital payments continues to rise in popularity, it also becomes susceptible to various malicious attacks. Successfully detecting anomalies within Blockchain transactions is essential for bolstering trust in digital payments. However, the task of anomaly detection in Blockchain transaction data is challenging due to the infrequent occurrence of illicit transactions. Although several studies have been conducted in the field, a limitation persists: the lack of explanations for the model's predictions. This study seeks to overcome this limitation by integrating eXplainable Artificial Intelligence (XAI) techniques and anomaly rules into tree-based ensemble classifiers for detecting anomalous Bitcoin transactions. The Shapley Additive exPlanation (SHAP) method is employed to measure the contribution of each feature, and it is compatible with ensemble models. Moreover, we present rules for interpreting whether a Bitcoin transaction is anomalous or not. Additionally, we have introduced an under-sampling algorithm named XGBCLUS, designed to balance anomalous and non-anomalous transaction data. This algorithm is compared against other commonly used under-sampling and over-sampling techniques. Finally, the outcomes of various tree-based single classifiers are compared with those of stacking and voting ensemble classifiers. Our experimental results demonstrate that: (i) XGBCLUS enhances TPR and ROC-AUC scores compared to state-of-the-art under-sampling and over-sampling techniques, and (ii) our proposed ensemble classifiers outperform traditional single tree-based machine learning classifiers in terms of accuracy, TPR, and FPR scores.

cs.LG↗

arXiv2023

Quantum tunneling from a new type of generalized Smith-Volterra-Cantor potential

In this paper, we introduce and analyze the Smith-Volterra-Cantor potential of power $ n $, denoted as SVC$\left(\rho, n\right)$. Bridging the gap between the general Cantor and SVC systems, this novel potential offers a fresh perspective on Cantor-like potential systems within quantum mechanics that unify fractal and non-fractal potentials. Utilizing the Super Periodic Potential (SPP) formalism, we derive the close form expression of the transmission probability $ T_{G}(k) $. Notably, the system exhibits exceptionally sharp transmission resonances, a characteristic that distinguishes it from other quantum systems. Furthermore, the multifaceted transmission attributes of the SVC$\left(\rho, n\right)$ are found to be critically dependent on both parameters, $ \rho $ and $ n $, offering an intricate interplay that warrants deeper exploration. Our findings highlight a pronounced scaling behavior of reflection probability with $ k $, which is underpinned by analytical derivations.

quant-ph↗

arXiv2023

Quantum tunneling from a new type of Unified Cantor Potential

We introduce a new type of potential system that combines the families of general Cantor (fractal system) and general Smith-Volterra-Cantor (non-fractal system) potentials. We call this system as Unified Cantor Potential (UCP) system. The UCP system of total span $L$ is characterized by scaling parameter $\rho >1$, stage $G$ and two real numbers $\alpha$ and $\beta$. For $\alpha=1$, $\beta=0$, the UCP system represents general Cantor potential while for $\alpha=0$, $\beta=1$, this system represent general Smith-Volterra-Cantor (SVC) potential. We provide close-form expression of transmission probability from UCP system for arbitrary $\alpha$ and $\beta$ by using $q$-Pochhammer symbol. Several new features of scattering are reported for this system. The transmission probability $T_{G}(k)$ shows a scaling behavior with $k$ which is derived analytically for this potential. The proposed system also opens up the possibility for further generalization of new potential systems that encompass a large class of fractal and non-fractal systems. The analytical formulation of tunneling from this system would help to study the transmission feature at breaking threshold when a system transit from fractal to non-fractal domain.

quant-ph↗

arXiv2023

Tunneling from general Smith-Volterra-Cantor potential

We study the tunneling problem from general Smith-Volterra-Cantor (SVC) potential of finite length $L$ characterized by the scaling parameter $\rho$ and stage $G$. We show that the SVC($\rho$) potential of stage $G$ is the special case of super periodic potential (SPP) of order $G$. By using SPP formalism developed by us earlier, we provide the close form expression of tunneling probability $T_{G}(k)$ with the help of $q$-Pochhammer symbol. The profile of $T_{G}(k)$ with wave vector $k$ is found to saturate with increasing stage $G$. Very sharp transmission resonances are found to occur from this system which may find applications in the design of sharp transmission filters.

quant-ph↗

arXiv2022

Quantum tunneling from family of Cantor potentials in fractional quantum mechanics

We explore the features of non-relativistic quantum tunneling in space fractional quantum mechanics through a family of Cantor potentials. We consider two types of potentials: general Cantor and general Smith-Volterra-Cantor potential. The Cantor potential is an example of fractal potential while the Smith-Volterra-Cantor potential doesn't belong to the category of a fractal system. The present study brings for the first time, the study of quantum tunneling through fractal potential in fractional quantum mechanics. We report several new features of scattering in the domain of space fractional quantum mechanics including the emergence of energy-band like features from these systems and extremely sharp transmission features. Further the scaling relation of the scattering amplitude with wave vector $k$ is presented analytically for both types of potentials.

quant-ph↗

arXiv2022

An enhanced method of initial cluster center selection for K-means algorithm

Clustering is one of the widely used techniques to find out patterns from a dataset that can be applied in different applications or analyses. K-means, the most popular and simple clustering algorithm, might get trapped into local minima if not properly initialized and the initialization of this algorithm is done randomly. In this paper, we propose a novel approach to improve initial cluster selection for K-means algorithm. This algorithm is based on the fact that the initial centroids must be well separated from each other since the final clusters are separated groups in feature space. The Convex Hull algorithm facilitates the computing of the first two centroids and the remaining ones are selected according to the distance from previously selected centers. To ensure the selection of one center per cluster, we use the nearest neighbor technique. To check the robustness of our proposed algorithm, we consider several real-world datasets. We obtained only 7.33%, 7.90%, and 0% clustering error in Iris, Letter, and Ruspini data respectively which proves better performance than other existing systems. The results indicate that our proposed method outperforms the conventional K means approach by accelerating the computation when the number of clusters is greater than 2.

cs.LG↗

arXiv2021

Locally finite free space as limiting case of PT-symmetric medium

We explicitly prove that the transfer matrix of a finite layered $PT$-symmetric system of fix length $L$ consisting of $N$ units of the potential system `$+iV$' and `$-iV$' of equal thickness becomes a unit matrix in the limit $N \rightarrow \infty$. This result is true for waves of arbitrary wave vector $k$. This shows that in this limit, the transmission coefficient is always unity while the reflection amplitude is zero for all waves traversing this length $L$. Therefore, a free space of finite length $L$ can be represented as a $PT$-symmetric medium.

quant-ph↗

arXiv2020

General(ized) Hartman effect

In this letter we prove explicitly that if Hartman effect exists for an arbitrary `unit cell' potential, then it also exist for a periodic system constructed using the same `unit cell' potential repeatedly. We further show that if Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick `unit cell' potential is same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive `unit cell' of the periodic system. Thus generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect.

quant-ph↗

arXiv2020

Role of $PT$-symmetry in understanding Hartman effect

The celebrated Hartman effect, according to which, the tunneling time through a opaque barrier is independent of the width of the barrier for a sufficiently thick barrier, is not well understood theoretically and experimentally till today. In this work we attempt to through some light to understand the mystery behind this paradoxical result of tunneling .For this purpose we calculate the tunneling time from a layered non-Hermitian system to examine the effect of $PT$-symmetry over tunneling time. We explicitly find that for system respecting $PT$-symmetry, the tunneling time saturates with the thickness of the $PT$-symmetric barrier and thus shows the existence of Hartman effect. For non PT-symmetric case, the tunneling time depends upon the thickness of the barrier and Hartman effect is lost. We further consider the limiting case in which the non-Hermitian system reduces to the real barrier to show that the Hartman effect from a real barrier is due to $PT$-symmetry (of the corresponding non-Hermitian system) .

quant-ph↗

arXiv2019

Self-similar features around spectral singularity in complex barrier potential

The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential $iV$, $V \in R^{+}$. We show that these loci are elliptical and self-similar to each other in $E-V$ plane where $E$ is energy of the wave. The orientations of these ellipses for reflection ($R$) and transmission ($T$) amplitude around a given SS are same. This is surprising due to the different mathematical forms of $R$ and $T$. In the vicinity of SS, $R \approx T$.

quant-ph↗

arXiv2019

Hartman effect from layered $PT$-symmetric system

The time taken by a wave packet to cross through a finite layered $PT$-symmetric system is calculated by stationary phase method. We consider the $PT$- symmetric system of fix spatial length $L$ consisting of $N$ units of the potential system `$+iV$' and `$-iV$' of equal width `$b$' such that $L=2Nb$. In the limit of large `$b$', the tunneling time is found to be independent of $L$ and therefore the layered $PT$-symmetric system display the Hartman effect. The interesting limit of $N \rightarrow \infty$ such that $L$ remains finite is investigated analytically. In this limit the tunneling time matches with the time taken to cross an empty space of length $L$. The result of this limiting case $N \rightarrow \infty$ also shows the consistency of phase space method of calculating the tunneling time despite the existence of controversial Hartman effect. The reason of Hartman effect is unknown to present day however the other definitions of tunneling time that indicate a delay which depends upon the length of traversing region have been effectively ruled out by recent attosecond measurements.

quant-ph↗

arXiv2019

Tunneling time from locally periodic potential in space fractional quantum mechanics

We calculate the time taken by a wave packet to travel through a classically forbidden locally periodic rectangular potential in space fractional quantum mechanics (SFQM). We obtain the close form expression of tunneling time from such a potential by stationary phase method. We show that tunneling time depends upon the width $b$ of the single barrier and separation $L$ between the barriers in the limit $b \to \infty$ and therefore generalized Hartman effect doesn't exist in SFQM. We observe that in SFQM, the tunneling time for large $b$ in the case of locally periodic potential is smaller than the tunneling from a single barrier of the same width $b$. It is further shown that with the increase in barrier numbers, the tunneling time reduces in SFQM in the limit of large $b$.

quant-ph↗

arXiv2019

New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics

The spectral singularity have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities have not been studied for the case of quaternionic potential. In the present work we perform an analytical study for the scattering from a quaternionic point interaction represented by delta function. New features of spectral singularities are observed which are different than the case of complex (non-quaternionic) point interaction. Most notable difference is the occurrence of spectral singularity from lossy point interaction which is forbidden in the case of standard non-Hermitian quantum mechanics.

quant-ph↗

arXiv2017

New scattering features in non-Hermitian space fractional quantum mechanics

The spectral singularity (SS) and coherent perfect absorption (CPA) have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian standard quantum mechanics (SQM) governed by Schrodinger equation. In the present work we explore these scattering features in the domain of non-Hermitian space fractional quantum mechanics (SFQM) governed by fractional Schrodinger equation which is characterized by Levy index $α$ ($ 1< α\leq 2$). We observe that non-Hermitian SFQM systems have more flexibility for SS and CPA and display some new features of scattering. For the delta potential $V(x)=-iρδ(x-x_{0})$, $ρ> 0$, the SS energy, $E_{ss}$, is blue or red shifted with decreasing $α$ depending the strength of the potential. For complex rectangular barrier in non-Hermitian SQM, it is known that the reflection and transmission amplitudes are oscillatory near the spectral singular point. It is found that these oscillations eventually develop SS in non-Hermitian SFQM. The similar features is also reported for the case of CPA phenomena from complex rectangular barrier in non-Hermitian SFQM. These observations suggest a deeper relation between scattering features of non-Hermitian SQM and non-Hermitian SFQM.

quant-ph↗

arXiv2017

Tunneling time in space fractional quantum mechanics

We calculate the time taken by a wave packet to travel through a classically forbidden region of space in space fractional quantum mechanics. We obtain the close form expression of tunneling time from a rectangular barrier by stationary phase method. We show that tunneling time depends upon the width $b$ of the barrier for $b \to \infty$ and therefore Hartman effect doesn't exist in space fractional quantum mechanics. Interestingly we found that the tunneling time monotonically reduces with increasing $b$. The tunneling time is smaller in space fractional quantum mechanics as compared to the case of standard quantum mechanics. We recover the Hartman effect of standard quantum mechanics as a special case of space fractional quantum mechanics.

quant-ph↗

arXiv2016

Broad Band X-ray Spectra of Atoll Source 4U 1636-536: NuSTAR and Swift Results

In this work we investigate broad band (1-79 keV) spectral nature of the atoll source 4U 1636-536 using the combined Nu-STAR and SWIFT-XRT data. The spectra are complex and have emission components from the disc, boundary-layer and corona. In addition to that a broad iron line is also observed. A relativistic line model assuming Schwarzchild metric fits this feature. The total flux varies from 1.4 $\times$ 10$^{-9}$ to 4.36 $\times$ 10$^{-9}$ $ergs/s/cm^2$. At the highest flux level the source was found in the soft state. In this state the Comptonized component has temperature $kT_e \sim 3$ keV and optical depth $τ\sim $ 16. We also detect a non-thermal tail with index $\sim$ 2.4, contributing $\sim$ 10 \% of the total flux in the soft state. We also find that the inner disc radius, electron temperature and optical depth vary with the total 0.1-100 keV unabsorbed flux. We discuss the implication of the results in this paper.

astro-ph.HE↗