On the Spectrum of Schrödinger Operators at the Half Space with a Certain Class of Boundary Conditions

1988 ◽  
Vol 7 (3) ◽  
pp. 233-239 ◽  
Author(s):  
Manfred Schröder
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Schrödinger Operators ◽  
Schrodinger Operators

The eigenvalue gap for one-dimensional Schrödinger operators with symmetric potentials

2009 ◽  
Vol 139 (2) ◽  
pp. 359-366 ◽  
Author(s):  
Min-Jei Huang ◽  
Tzong-Mo Tsai
Keyword(s):  
Boundary Conditions ◽  
Schrödinger Operators ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Eigenvalue Gap

We consider the eigenvalue gap for Schrödinger operators on an interval with Dirichlet or Neumann boundary conditions. For a class of symmetric potentials, we prove that the gap between the two lowest eigenvalues is maximized when the potential is constant. We also give some related results for doubly symmetric potentials.

NORM RESOLVENT CONVERGENCE TO MAGNETIC SCHRÖDINGER OPERATORS WITH POINT INTERACTIONS

2001 ◽  
Vol 13 (04) ◽  
pp. 465-511 ◽  
Author(s):  
HIDEO TAMURA
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Two Dimensions ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Point Interactions ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence

The Schrödinger operator with δ-like magnetic field at the origin in two dimensions is not essentially self-adjoint. It has the deficiency indices (2, 2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We here consider Schrödinger operators with magnetic fields of small support and study the norm resolvent convergence to Schrödinger operator with δ-like magnetic field. We are concerned with the boundary conditions realized in the limit when the support shrinks. The results obtained heavily depend on the total flux of magnetic field and on the resonance space at zero energy, and the proof is based on the analysis at low energy for resolvents of Schrödinger operators with magnetic potentials slowly falling off at infinity.

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