TY - JOUR
T1 - On the spectra of one-dimensional schrödinger operators with singular potentials
AU - Rabinovich, Vladimir S.
AU - Barrera-Figueroa, Víctor
AU - Ramírez, Leticia Olivera
N1 - Publisher Copyright:
© 2019 Rabinovich, Barrera-Figueroa and Olivera Ramírez.
PY - 2019
Y1 - 2019
N2 - The paper is devoted to the spectral properties of one-dimensional Schrödinger operators Squ(x)=(-d2dx2+q(x))u(x), x∈ℝ, (1) with potentials q = qo+qs, where qo∈L∞(ℝ) is a regular potential, and qs∈D'(ℝ) is a singular potential with support on a discrete infinite set Y⊂ℝ. We consider the extension H of formal operator (1) to an unbounded operator in L2(ℝ) defined by the Schrödinger operator Sqo with regular potential qo and interaction conditions at the points of the set Y. We study the closedness and self-adjointness of H. If the set Y≃Z has a periodic structure we give the description of the essential spectrum of operator H in terms of limit operators. For periodic potentials qo we consider the Floquet theory of H, and apply the spectral parameter power series method for determining the band-gap structure of the spectrum. We also consider the case when the regular periodic part of the potential is perturbed by a slowly oscillating at infinity term. We show that this perturbation changes the structure of the spectra of periodic operators significantly. This works presents several numerical examples to demonstrate the effectiveness of our approach.
AB - The paper is devoted to the spectral properties of one-dimensional Schrödinger operators Squ(x)=(-d2dx2+q(x))u(x), x∈ℝ, (1) with potentials q = qo+qs, where qo∈L∞(ℝ) is a regular potential, and qs∈D'(ℝ) is a singular potential with support on a discrete infinite set Y⊂ℝ. We consider the extension H of formal operator (1) to an unbounded operator in L2(ℝ) defined by the Schrödinger operator Sqo with regular potential qo and interaction conditions at the points of the set Y. We study the closedness and self-adjointness of H. If the set Y≃Z has a periodic structure we give the description of the essential spectrum of operator H in terms of limit operators. For periodic potentials qo we consider the Floquet theory of H, and apply the spectral parameter power series method for determining the band-gap structure of the spectrum. We also consider the case when the regular periodic part of the potential is perturbed by a slowly oscillating at infinity term. We show that this perturbation changes the structure of the spectra of periodic operators significantly. This works presents several numerical examples to demonstrate the effectiveness of our approach.
KW - Dispersion equation
KW - Limit operators method
KW - Monodromy matrices
KW - Periodic Schrödinger operators
KW - Slowly oscillating at infinity perturbation
KW - Spectral parameter power series (SPPS) method
UR - http://www.scopus.com/inward/record.url?scp=85065404679&partnerID=8YFLogxK
U2 - 10.3389/fphy.2019.00057
DO - 10.3389/fphy.2019.00057
M3 - Artículo
AN - SCOPUS:85065404679
SN - 2296-424X
VL - 7
JO - Frontiers in Physics
JF - Frontiers in Physics
IS - MAR
M1 - 57
ER -