Is this project an undergraduate, graduate, or faculty project?
Faculty
Project Type
individual
Campus
Daytona Beach
Authors' Class Standing
Faculty
Lead Presenter's Name
Keshav Acharya
Faculty Mentor Name
N/A
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Abstract
In this research, we discuss some important properties of half line Titcchmarsh-Weyl m functions associated to the vector-valued discrete Schrodinger operators induced by the second order difference expression. The Titchmarsh-Weyl m functions provide explicit description of absolutely continuous, singular continuous and pure point spectrum of corresponding Schrodinger operators. The Remling's theorem utilizes these m functions to describe the absolutely continuous spectrum. We have established that these m functions are matrix-valued Herglotz functions maping complex upper half plane to Siegel Space, a generalization of complex upper half plane. We then define a metric on the Siegel space as a generalization of the hyperbolic metric on complex upper half plane. Then we establish the distance decreasing property of these m functions with respect to the metric we defined, for vector-valued discrete Schrodinger operators. This property of functions is essential to prove the Remling's theorem.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Half Line Titchmarsh Weyl m functions of vector-valued discrete Schrodinger operators
In this research, we discuss some important properties of half line Titcchmarsh-Weyl m functions associated to the vector-valued discrete Schrodinger operators induced by the second order difference expression. The Titchmarsh-Weyl m functions provide explicit description of absolutely continuous, singular continuous and pure point spectrum of corresponding Schrodinger operators. The Remling's theorem utilizes these m functions to describe the absolutely continuous spectrum. We have established that these m functions are matrix-valued Herglotz functions maping complex upper half plane to Siegel Space, a generalization of complex upper half plane. We then define a metric on the Siegel space as a generalization of the hyperbolic metric on complex upper half plane. Then we establish the distance decreasing property of these m functions with respect to the metric we defined, for vector-valued discrete Schrodinger operators. This property of functions is essential to prove the Remling's theorem.