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Abstract <p>In 2000, A.M. Savchuk and A.A. Shkalikov showed that the regularized trace of the Sturm–Liouville operator with a potential in the form of a distribution is a nonlinear functional of the potential. It turned out that the linearity or nonlinearity of the regularized traces of the Sturm–Liouville operator depends on the type of boundary conditions. In the Dirichlet case, the regularized trace is a nonlinear functional, and in the case of periodic boundary conditions, the nonlinearity of the regularized trace is violated. In this paper, we show that the regularized traces of the Sturm–Liouville operator, both in the case of Robin conditions and in the case of an analogue of periodic conditions on a star graph, are nonlinear functionals of the initial data of the operator.</p>

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On Nonlinear Regularized Traces of Second-Order Differential Operators on a Star Graph

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • Z. Yu. Fazullin,
  • B. E. Kanguzhin,
  • Z. Z. Satpaeva

摘要

Abstract

In 2000, A.M. Savchuk and A.A. Shkalikov showed that the regularized trace of the Sturm–Liouville operator with a potential in the form of a distribution is a nonlinear functional of the potential. It turned out that the linearity or nonlinearity of the regularized traces of the Sturm–Liouville operator depends on the type of boundary conditions. In the Dirichlet case, the regularized trace is a nonlinear functional, and in the case of periodic boundary conditions, the nonlinearity of the regularized trace is violated. In this paper, we show that the regularized traces of the Sturm–Liouville operator, both in the case of Robin conditions and in the case of an analogue of periodic conditions on a star graph, are nonlinear functionals of the initial data of the operator.