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

• URL: http://arxiv.org/abs/2308.05195v2
• Date: Fri, 13 Oct 2023 00:36:24 GMT
• Title: 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
• Authors: Michael Maroun
• Abstract summary: This work deals with the point spectrum, i.e. bound state energy eigenvalue, of the Dirac delta function in two and three dimensions. Due to the uniqueness of the solution presented here, it is immediate that the linear operator ensures that the point spectrum has exactly one element.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Analyzing the point spectrum, i.e. bound state energy eigenvalue, of the Dirac delta function in two and three dimensions is notoriously difficult without recourse to regularization or renormalization, typically both. The reason for this in two dimensions is two fold; 1) the coupling constant, together with the mass and Planck's constant form an unitless quantity. This causes there to be a missing anomalous length scale. 2) The immediately obvious L$^2$ solution is divergent at the origin, where the Dirac Delta potential has its important point of support as a measure. Due to the uniqueness of the solution presented here, it is immediate that the linear operator (the two dimensional Laplace operator on all of $\mathbb{R}^2$), with the specialized domain constructed here, ensures that the point spectrum has exactly one element. This element is determined precisely, and a natural mathematically rigorous resolution to the anomalous length scale arises. In this work, there is no recourse to renormalization or regularization of any kind.

Related papers

• Exact dynamics of quantum dissipative $XX$ models: Wannier-Stark localization in the fragmented operator space [49.1574468325115]
  We find an exceptional point at a critical dissipation strength that separates oscillating and non-oscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace.
  arXiv  Detail & Related papers  (2024-05-27T16:11:39Z)
• Renormalization group and spectra of the generalized P\"oschl-Teller potential [0.0]
  We study the P"oschl-Teller potential $V(x) = alpha2 g_s sinh-2(alpha x) + alpha2 g_c cosh-2(alpha x)$, for every value of the dimensionless parameters $g_s$ and $g_c singularity. We show that supersymmetry of the potential, when present, is also spontaneously broken, along with conformal symmetry.
  arXiv  Detail & Related papers  (2023-08-08T21:44:55Z)
• Small-time controllability for the nonlinear Schr\"odinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields [55.2480439325792]
  We address the small-time controllability problem for a nonlinear Schr"odinger equation (NLS) on $mathbbRN$ in the presence of magnetic and electric external fields. In detail, we study when it is possible to control the dynamics of (NLS) as fast as desired via sufficiently large control signals.
  arXiv  Detail & Related papers  (2023-07-28T21:30:44Z)
• Algebraic discrete quantum harmonic oscillator with dynamic resolution scaling [22.20907440445493]
  We develop an algebraic formulation for the discrete quantum harmonic oscillator (DQHO) This formulation does not depend on the discretization of the Schr"odinger equation and recurrence relations of special functions. The coherent state of the DQHO is constructed, and its expected position is proven to oscillate as a classical harmonic oscillator.
  arXiv  Detail & Related papers  (2023-04-04T03:02:03Z)
• Global Convergence Rate of Deep Equilibrium Models with General Activations [18.601449856300984]
  This paper shows that the fact still holds for DEQs with any generally bounded activation with bounded first and second derivatives. Since the new activation function is generally non-homogeneous, bounding the least eigenvalue of the Gram matrix of the equilibrium point is particularly challenging.
  arXiv  Detail & Related papers  (2023-02-11T22:18:26Z)
• Reduced Density Matrices and Moduli of Many-Body Eigenstates [1.261852738790008]
  eigenstate moduli problem is closely related to the $N$-representability conditions for 2-reduced density matrices. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature.
  arXiv  Detail & Related papers  (2023-01-04T03:14:07Z)
• Spectral asymptotics for two-dimensional Dirac operators in thin waveguides [0.0]
  We prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schr"odinger operator on $L1(mathbb R)$ [ mathcalL_e := -fracd2ds2 - frackappa2pi2 ] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $varepsilon$ from the essential spectrum, where $2varepsilon$ is the width
  arXiv  Detail & Related papers  (2022-07-18T15:50:10Z)
• Superdiffusion in random two dimensional system with time-reversal symmetry and long-range hopping [45.873301228345696]
  localization problem in the crossover regime for the dimension $d=2$ and hopping $V(r) propto r-2$ is not resolved yet. We show that for the hopping determined by two-dimensional anisotropic dipole-dipole interactions there exist two distinguishable phases at weak and strong disorder.
  arXiv  Detail & Related papers  (2022-05-29T16:53:20Z)
• The Energy Eigenvalue for the Singular Wave Function of the Three Dimensional Dirac Delta Schrodinger Potential via Distributionally Generalized Quantum Mechanics [0.0]
  Indeterminacy originates from a lack of scale. Wave function not being well defined at the support of the generalized function. Problem is solved here in a completely mathematically rigorous manner.
  arXiv  Detail & Related papers  (2021-01-18T09:07:31Z)
• The hydrogen atom: consideration of the electron self-field [0.0]
  We substantiate the need for account of the proper electromagnetic field of the electron in the canonical problem of hydrogen. We demonstrate that only particular classes of solutions, "nonlinear" analogues of s- and p-states, can be obtained through decomposition of a solution in a series.
  arXiv  Detail & Related papers  (2021-01-05T21:52:41Z)
• Anisotropy-mediated reentrant localization [62.997667081978825]
  We consider a 2d dipolar system, $d=2$, with the generalized dipole-dipole interaction $sim r-a$, and the power $a$ controlled experimentally in trapped-ion or Rydberg-atom systems. We show that the spatially homogeneous tilt $beta$ of the dipoles giving rise to the anisotropic dipole exchange leads to the non-trivial reentrant localization beyond the locator expansion.
  arXiv  Detail & Related papers  (2020-01-31T19:00:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.