Abstract
The Cauchy problem is considered for the Hartree equation with a delta potential in three dimensions. We prove existence and uniqueness of solutions in a Colombeau type algebra. This algebra is a quotient space and we show that certain classical spaces are embedded in it. In this manner the product \(\delta \cdot u\) obtains a meaning. We observe a regularized equation and obtain estimates for the \(H^2\) norm of the solution. The delta function is regularized by a mollifier. Hence the net of solutions of the regularized equation gives rise to an element of the Colombeau algebra. On the other hand, there is a well–posedness result for this type of equation in the singular Sobolev space, the domain of the fractional Laplacian. We compare the two different settings and prove a compatibility result.
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We are thankful to Professor Marko Nedeljkov for his feedback and useful discussions and ideas.
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Communicated by Michael Kunzinger.
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The authors acknowledge financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-9/2021-14/200125). The second author acknowledges financial support of the Croatian Science Foundation under the project 2449 MiTPDE
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Dugandžija, N., Vojnović, I. Singular solution of the Hartree equation with a delta potential. Monatsh Math 200, 799–818 (2023). https://doi.org/10.1007/s00605-022-01804-z
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DOI: https://doi.org/10.1007/s00605-022-01804-z