A Second Regularized Trace Formula for a 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.

The formula for the regularized trace of the Sturm-Liouville operator with a logarithmic potential

We have obtained a regularized trace formula for the Sturm-Liouville operator on a semi-axis with a logarithmic potential.

Spectral properties of the iterated Laplacian with a potential in a punctured domain

In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order m 2 N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ? d ? 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called ?nonharmonic? analysis.