Bound states for multiple Dirac-δ wells in space-fractional quantum mechanics

Abstract

A novel application of a Fourier integral representation of bound states in quantum mechanics Am.Using the momentum-space approach, we obtain bound states for multiple Dirac-δ wells in the framework of space-fractional quantum mechanics. Introducing first an attractive Dirac-comb potential, i.e., Dirac comb with strength − g (g > 0), in the spacefractional Schrödinger equation we show that the problem of obtaining eigenenergies of a system with N Dirac-δ wells can be reduced to a problem of obtaining the eigenvalues of an N × N matrix. As an illustration we use the present matrix formulation to derive expressions satisfied by the bound-state energies of N = 1, 2, 3 delta wells. We also obtain the corresponding wave functions and express them in terms of Fox's H-function. C 2014 AIP Publishing LLC. [http://dx.

Fractional powers and singular perturbations of quantum differential Hamiltonians

“…A relevant example are the powers of the quantum-mechanical semi-relativistic energy operator √ −∆ + m 2 , the singular perturbation of which yields precisely operators of the type d (s/2) m 2 ,τ considered in Section 6 or, in the case of zero rest energy, of the type k (s/2) τ as in Section 4. What one finds in the literature is an increasing amount of recent studies [14,16,4,11,18,20,10,15] where the singular perturbation of the fractional Laplacian is approached through Green's function methods (together with Wick-like rotations to obtain the propagator from the resolvent) that have the virtue of highlighting the singular structure carried over by what we denoted with G s,λ and G s,λ , however with no specific concern to the multiplicity of self-adjoint realisations and the associated local boundary conditions, or to the increase of the deficiency index with the power s.…”

“…Recently, especially for the solution theory of non-linear Schrödinger equations whose linear part is governed by singular Hamiltonians of point interactions [8,12], as well as for linear Schrödinger-like equations with singular perturbations of fractional powers of the Laplacian [14,16,4,11,18,20,10,15,17], the interest has increased around various ways of combining the two constructions above.…”

Point-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range

“…In the last decade an amount of studies focused, in particular in application to the context of fractional quantum mechanics, on linear Schrödinger equations governed by the linear operator (1.1) (−∆) s/2 + singular perturbation at x 0 for some fixed point x 0 ∈ R d and some s > 0, that is, Schrödinger equations for a singular perturbation of a fractional power of the Laplacian [15,17,6,13,19,21,9,16,18]. Motivated by that, in a recent work in collaboration with A. Ottolini [14] we set up the systematic construction and classification of all the self-adjoint realisations in L 2 (R d ) of the operators of the form (1.1) through a natural 'restriction-extension' procedure: first one restricts the operator (−∆) s/2 (initially defined, e.g., as a Fourier multiplier) to smooth functions vanishing in neighbourhoods of x 0 , and then one builds all the operator extensions of such restriction that are self-adjoint on L 2 (R d ).…”

Confining massless Dirac particles in two-dimensional curved space

“…2-dimensional systems are especially of interest due to graphene, 2D topological insulators and the fractional quantum Hall effect 16 . Research is focused on bound states of Dirac particles for example in the context of quantum computing and waveguides 17,18 .…”