## 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.