2017/02/23 by A. V. Zolotaryuk · 1 citation
Mathematics · Physics and Astronomy · #Complex plane #Constant (computer programming) #Countable set #Geometry #Limit (mathematics) #Mathematical analysis #Mathematics #Opacity #Physics #Piecewise #Planar #Plane (geometry) #Point (geometry) #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Resonance (particle physics) #Singular point of a curve #Singularity #Spectral Theory in Mathematical Physics #Zero (linguistics) #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/1751-8121/aa6dc2
5 figures. Significantly modified version of arXiv:1610.07288
arxiv created 2017/02/23 · openalex publication_date 2017/05/09 · arxiv updated 2017/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract Several families of one-point interactions are derived from the system consisting of two and three δ -potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of δ -approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mstyle> </mml:math> , so that the intensity of each δ -potential is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msup> </mml:mstyle> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> , j = 1, 2, 3, the width of each layer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>l</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ε</mml:mi> </mml:mstyle> </mml:math> and the distance between the layers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>ε</mml:mi> <mml:mi>τ</mml:mi> </mml:msup> </mml:mstyle> </mml:math> , c > 0. It is shown that at some values of the intensities a 1 , a 2 and a 3 , the transmission across the limit point potentials is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. Within the interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>μ</mml:mi> <mml:mo><</mml:mo> <mml:mn>2</mml:mn> </mml:mstyle> </mml:math> , the resonance sets consist of two curves on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> -plane and three surfaces in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> -space. As the parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>μ</mml:mi> </mml:mstyle> </mml:math> approaches the value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mstyle> </mml:math> , three types of splitting the one-point interactions into countable families are observed.