In this paper, we consider the one-dimensional semirelativistic Schrödinger equation for a particle interacting with $N$ Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an $N \times N$ matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most $N$ bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a self-adjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for $N$ centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two-center case. We observe the so-called threshold anomaly for two symmetrically located centers. The semirelativistic version of the Kronig-Penney model is shortly discussed, and the band gap structure of the spectrum is illustrated. The bound state and scattering problems in the massless case are also discussed. Furthermore, the reflection and the transmission coefficients for the two delta potentials in this particular case are analytically found. Finally, we solve the renormalization group equations and compute the beta function nonperturbatively.
In this manuscript, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.
Motivated from studies on anomalous diffusion, we show that the memory function $M(t)$ of complex materials, that their creep compliance follows a power law, $J(t)\sim t^q$ with $q\in \mathbb{R}^+$, is the fractional derivative of the Dirac delta function, $\frac{\mathrm{d}^qδ(t-0)}{\mathrm{d}t^q}$ with $q\in \mathbb{R}^+$. This leads to the finding that the inverse Laplace transform of $s^q$ for any $q\in \mathbb{R}^+$ is the fractional derivative of the Dirac delta function, $\frac{\mathrm{d}^qδ(t-0)}{\mathrm{d}t^q}$. This result, in association with the convolution theorem, makes possible t…
The Energy Eigenvalue for the Singular Wave Function of the Three Dimensional Dirac Delta Schrodinger Potential via Distributionally Generalized Quantum Mechanics
Isospectral potentials with Dirac delta interaction: Constrained Spectra
Mahidol University（马希托大学）；Maharaja Sriram Chandra Bhanja Deo University（Maharaja Sriram Chandra Bhanja Deo 大学）；Indian Institute of Science Education and Research Kolkata（加尔各答印度科学教育与研究学院）；Siksha O Anusandhan
The Exact Point Spectrum and Eigenvector of the Unique Continuous L$^2(\mathbb{R}^2)$ Bound State Solution to the Dirac Delta Schrodinger Potential in Two Dimensions
