Spectral properties of the massless relativistic harmonic oscillator

Lőrinczi, József; Małecki, Jacek

doi:10.1016/j.jde.2012.07.010

Cited by 52 publications

(59 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As well, no its analog is known in the current context. Nonetheless, our simulation routines will reproduce the standard nodal picture for approximate eigenvectors, in consistency with previously established analytic properties of the Cauchy oscillator eigenfunctions, [24,25]. (8), for a = 50, 100, 200, 500.…”

Section: Remarksupporting

confidence: 53%

“…To test a predictive power of the just outlined computer-assisted method of solution of the Schrödinger-type spectral problem, while extended to a non-local operator H, we shall take advantage of the existence in L 2 (R) of a complete analytic solution of the Cauchy oscillator problem, [1,25].…”

Section: Cauchy Oscillator: Strang Methods Versus Exact Spectral mentioning

confidence: 99%

“…First we consider an analytically solvable Cauchy oscillator problem, [24][25][26][27], which is viewed as a test model for an analysis of possible intricacies/pitfalls of adopted numerical procedures. Next we shall pass to the Cauchy-Schrödinger well problem.…”

Section: Remarkmentioning

confidence: 99%

“…(2) is that the integral is interpreted in terms of its Cauchy principal value. Clearly, H needs to be considered as a self-adjoint operator and its spectrum is expected to belong to R + , [24,25]. Then, the one-parameter family {e −tH , t 0} constitutes a (strongly continuous symmetric) semigroup of interest.…”

Section: Solution Of the Schrödinger Eigenvalue Problem By Means mentioning

confidence: 99%

“…We display the k-evolution, up to k = 3000, of E (k) i for selected indices i = 3, 4, 5, and for values a = 50, 100, 200 of the boundary parameter. We also depict the approximations (5) of the corresponding eigenfunctions ψ i , i = 3, 4, 5 of the Cauchy oscillator, [24,25]. We note that the plots (e.g.…”

Section: N ≥mentioning

confidence: 99%

See 4 more Smart Citations

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

Żaba

Garbaczewski

2014

Journal of Mathematical Physics

111

View full text Add to dashboard Cite

This paper is a direct offspring of Ref. [1] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions was maid with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e. detailed eigenvalues and shapes of eigenfunctions, has been analyzed.2

show abstract

Section: Remarksupporting

confidence: 53%

Section: Cauchy Oscillator: Strang Methods Versus Exact Spectral mentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Solution Of the Schrödinger Eigenvalue Problem By Means mentioning

confidence: 99%

Section: N ≥mentioning

confidence: 99%

See 3 more Smart Citations

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

Żaba

Garbaczewski

2014

Journal of Mathematical Physics

111

View full text Add to dashboard Cite

show abstract

Bulk Behaviour of Ground States for Relativistic Schrödinger Operators with Compactly Supported Potentials

Ascione,

Lőrinczi

2023

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

We propose a probabilistic representation of the ground states of massive and massless Schrödinger operators with a potential well in which the behaviour inside the well is described in terms of the moment-generating function of the first exit time from the well and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part, we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schrödinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.

show abstract

Extension technique for complete Bernstein functions of the Laplace operator

Kwaśnicki

Mucha

2018

J. Evol. Equ.

View full text Add to dashboard Cite

We discuss representation of certain functions of the Laplace operator ∆ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies (−∆) 1/2 , the square root of the d-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the (d + 1)-dimensional Laplace operator ∆ t,x in (0, ∞) × R d . Caffarelli and Silvestre extended this to fractional powers (−∆) α/2 , which correspond to operators ∇ t,x (t 1−α ∇ t,x ). We provide an analogous result for all complete Bernstein functions of −∆ using Krein's spectral theory of strings.Two sample applications are provided: a Courant-Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrödinger operators ψ(−∆) + V (x), as well as an upper bound for the eigenvalues of these operators. Here ψ is a complete Bernstein function and V is a confining potential.

show abstract

Spectral properties of the massless relativistic harmonic oscillator

Cited by 52 publications

References 22 publications

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

Bulk Behaviour of Ground States for Relativistic Schrödinger Operators with Compactly Supported Potentials

Extension technique for complete Bernstein functions of the Laplace operator

Contact Info

Product

Resources

About