On the Degenerate Spectral Gaps of the One-Dimensional Schrödinger Operators with Periodic Point Interactions

2012 ◽  
Vol 44 (4) ◽  
pp. 2847-2870
Author(s):  
Hiroaki Niikuni
Keyword(s):  
Periodic Point ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Spectral Gaps ◽  
Point Interactions ◽  
One Dimensional ◽  
The One

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

2001 ◽  
Vol 64 (1) ◽  
pp. 125-143 ◽  
Author(s):  
MEIRONG ZHANG
Keyword(s):  
Dirichlet Problem ◽  
Rotation Number ◽  
Periodic Potential ◽  
Schrödinger Operators ◽  
Negative Integer ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Periodic Potentials ◽  
The One

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

scholarly journals Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials

2013 ◽  
Vol 143 (4) ◽  
pp. 791-816 ◽  
Author(s):  
Yu. D. Golovaty ◽  
R. O. Hryniv
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
One Dimensional ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence ◽  
The One ◽  
Image Position

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

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