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The weakly coupled fractional one-dimensional Schrödinger operator with index 1 < α ≤ 2

### Abstract

Considering the space fractional Weyl operator on the separable Hilbert space we determine the asymptotic behavior of both the free Green's function and its variation with respect to energy in one dimension for bound states. Later, we specify the Birman–Schwinger representation for the Schrödinger operator and extract the finite-rank portion which is essential for the asymptotic expansion of the ground state. Finally, we determine necessary and sufficient conditions for there to be a bound state for small coupling constant *g*.

© 2010 American Institute of Physics

Received 14 January 2010
Accepted 23 November 2010
Published online 30 December 2010

Acknowledgments:
The author is deeply grateful to the referees for their fruitful comments and suggestions.

Article outline:

I. INTRODUCTION
II. DEFINITION AND SELF-ADJOINTNESS OF THE FRACTIONAL WEYL OPERATOR
III. THE FREE GREEN'S FUNCTION AND ITS DERIVATIVE
IV. REPRESENTATION OF THE BIRMAN–SCHWINGER OPERATOR FOR THE SCHRODINGER OPERATOR
V. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE GROUND STATE ENERGY
VI. CONCLUSIONS