On the inverse transmission eigenvalue problem with a piecewise $W_2^1$ refractive index
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908558975565824 |
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| author | Liu, Tao Lyu, Kang Wei, Guangsheng Yang, Chuan-Fu |
| author_facet | Liu, Tao Lyu, Kang Wei, Guangsheng Yang, Chuan-Fu |
| contents | In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius $b$. Instead of the usual case of the refractive index $ρ\in W^2_2$, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $ρ$ is a piecewise $ W^1_2$ function. We prove that if $\int_0^b \sqrt{ρ(r)} dr<b$, then $ρ$ is uniquely determined by all special transmission eigenvalues; if $\int_0^b \sqrt{ρ(r)} dr=b$, then all special transmission eigenvalues with some additional information can uniquely determine $ρ$. We also consider the mixed spectral problem and obtain that $ρ$ is uniquely determined from partial information of $ρ$ and the ``almost real subspectrum". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the inverse transmission eigenvalue problem with a piecewise $W_2^1$ refractive index Liu, Tao Lyu, Kang Wei, Guangsheng Yang, Chuan-Fu Spectral Theory Mathematical Physics Primary 34A55, Secondary 34L40, 34L20 In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius $b$. Instead of the usual case of the refractive index $ρ\in W^2_2$, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $ρ$ is a piecewise $ W^1_2$ function. We prove that if $\int_0^b \sqrt{ρ(r)} dr<b$, then $ρ$ is uniquely determined by all special transmission eigenvalues; if $\int_0^b \sqrt{ρ(r)} dr=b$, then all special transmission eigenvalues with some additional information can uniquely determine $ρ$. We also consider the mixed spectral problem and obtain that $ρ$ is uniquely determined from partial information of $ρ$ and the ``almost real subspectrum". |
| title | On the inverse transmission eigenvalue problem with a piecewise $W_2^1$ refractive index |
| topic | Spectral Theory Mathematical Physics Primary 34A55, Secondary 34L40, 34L20 |
| url | https://arxiv.org/abs/2509.06520 |