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Schrödinger Operators with Oblique Transmission Conditions in .

Jussi Behrndt1, Markus Holzmann1, Georg Stenzel1

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This study explores Schrödinger operators with novel oblique transmission conditions. Attractive interactions lead to an unbounded discrete spectrum, differing significantly from standard conditions.

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Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Spectral Theory

Background:

  • Schrödinger operators are fundamental in quantum mechanics, describing the behavior of quantum systems.
  • Standard transmission conditions at boundaries are well-understood, but novel conditions offer new modeling possibilities.
  • The spectral properties of these operators dictate the possible energy states and dynamics of a quantum system.

Purpose of the Study:

  • To investigate the spectral properties of self-adjoint Schrödinger operators with oblique transmission conditions.
  • To analyze the impact of these new conditions on the discrete and essential spectrum.
  • To establish theoretical tools like Krein-type resolvent formula and Birman-Schwinger principle for these operators.

Main Methods:

  • Analysis of self-adjoint Schrödinger operators defined on a domain with a smooth closed curve boundary.
  • Application of oblique transmission conditions, utilizing the Wirtinger derivative instead of the normal derivative.
  • Spectral analysis techniques to determine discrete and essential spectrum, and derivation of resolvent formulas.

Main Results:

  • Demonstration that oblique transmission conditions lead to significantly different spectral properties compared to standard conditions.
  • Discovery that for attractive interactions, the discrete spectrum of these operators is unbounded below.
  • Identification of the essential spectrum and proof of a Krein-type resolvent formula and Birman-Schwinger principle.

Conclusions:

  • Schrödinger operators with oblique transmission conditions exhibit unique spectral behavior, particularly an unbounded discrete spectrum for attractive interactions.
  • These operators serve as valid models in quantum mechanics, arising as non-relativistic limits of Dirac operators with specific interactions.
  • The study provides a theoretical framework for understanding these novel operators and their physical implications.