A second regularized trace formula for a higher order differential operator

2020 ◽  
Author(s):  
Erdal Gül
Keyword(s):  
Differential Operator ◽  
Trace Formula ◽  
Higher Order ◽  
Order Differential Operator ◽  
Regularized Trace

scholarly journals A Second Regularized Trace Formula for a Fourth Order Differential Operator

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 629
Author(s):  
Erdal Gül ◽  
Aylan Ceyhan
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
State Space ◽  
Trace Formula ◽  
Fourth Order ◽  
The State ◽  
Order Differential Operator ◽  
Exact Nature ◽  
Regularized Trace ◽  
Fourth Order Differential Operator

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

scholarly journals A symmetrized conjugacy scheme for orthogonal expansions

2013 ◽  
Vol 143 (2) ◽  
pp. 427-443 ◽  
Author(s):  
Adam Nowak ◽  
Krzysztof Stempak
Keyword(s):  
Differential Operator ◽  
Higher Order ◽  
Riesz Transforms ◽  
Orthogonal System ◽  
Order Differential Operator ◽  
Orthogonal Expansions ◽  
Partial Derivatives ◽  
Poisson Integrals ◽  
Conjugate Poisson Integrals ◽  
Classical Shape

We establish a symmetrization procedure in the context of general orthogonal expansions associated with a second-order differential operator L, a Laplacian. Combining with a unified conjugacy scheme from an earlier paper by Nowak and Stempak permits, using a suitable embedding, a differential-difference Laplacian $\mathbb{L}$ to be associated with the initially given orthogonal system of eigenfunctions of L, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related partial derivatives decomposing $\mathbb{L}$ are skew-symmetric in an appropriate L2 space and they commute with Riesz transforms and conjugate Poisson integrals. The results also shed new light on the issue of defining higher-order Riesz transforms for general orthogonal expansions.

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