Advanced Search
Article Contents
Article Contents

Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Figure(4) Related Papers Cited by
  • We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q = -\Delta+q$ on a bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partial\Omega$. More precisely, for $\gamma$ an open and non-empty subset of $\partial\Omega$, we consider the boundary spectral data on $\gamma$ given by ${\rm BSD}(q, \gamma): = \{(\lambda_{k}, {\partial_\nu \varphi_{k}}_{|\gamma}):\ k \geq1\}$, where $\{ \lambda_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ \varphi_k:\ k \geq1 \}$ an associated orthonormal basis of eigenfunctions, and $\nu$ is the unit outward normal vector to $\partial\Omega$. Our main result consists of determining a bounded potential $q\in L^\infty(\Omega)$ from the data ${\rm BSD}(q, \gamma)$. Previous uniqueness results, with arbitrarily small $\gamma$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.

    Mathematics Subject Classification: 35R30, 35J10, 35L05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Geometric condition (3.10)

    Figure 2.  Sets $A_{x, \varepsilon}$

    Figure 3.  Support of the geometric optics solution

    Figure 4.   

  •   C. Bardos , G. Lebeau  and  J. Rauch , Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM Journal on Control and Optimization, 30 (1992) , 1024-1065.  doi: 10.1137/0330055.
      M. Belishev , An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987) , 524-527. 
      M. Belishev  and  Y. Kurylev , To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992) , 767-804.  doi: 10.1080/03605309208820863.
      G. Borg , Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946) , 1-96.  doi: 10.1007/BF02421600.
      B. Canuto and O. Kavian, Determining two coefficients in elliptic operators via boundary spectral data: A uniqueness result, Bolletino Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7 (2004), 207–230.
      M. Choulli  and  P. Stefanov , Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data, Commun. Partial Diff. Eqns., 38 (2013) , 455-476. 
      I. M. Gel'fand  and  B. M. Levitan , On the determination of a differential equation from its spectral function, Izv. Akad. Nauk USSR, Ser. Mat., 15 (1951) , 309-360. 
      L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976.
      H. Isozaki , Some remarks on the multi-dimensional Borg-Levinson theorem, J. Math. Kyoto Univ., 31 (1991) , 743-753.  doi: 10.1215/kjm/1250519727.
      A. Katchalov  and  Y. Kurylev , Multidimensional inverse problem with incomplete boundary spectral data, Commun. Partial Diff. Eqns., 23 (1998) , 55-95.  doi: 10.1080/03605309808821338.
      A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.
      A. Katchalov , Y. Kurylev , M. Lassas  and  N. Mandache , Equivalence of time-domain inverse problems and boundary spectral problem, Inverse problems, 20 (2004) , 419-436.  doi: 10.1088/0266-5611/20/2/007.
      O. Kavian , Y. Kian  and  E. Soccorsi , Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, Journal de Mathématiques Pures et Appliquées, 104 (2015) , 1160-1189.  doi: 10.1016/j.matpur.2015.09.002.
      Y. Kian , A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data, J. Spectr. Theory, 8 (2018) , 235-269.  doi: 10.4171/JST/195.
      Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, IMRN, 2017, https://doi.org/10.1093/imrn/rnx263. doi: 10.1093/imrn/rnx263.
      Y. Kurylev , An inverse boundary problem for the Schrödinger operator with magnetic field, Journal of Mathematical Physics, 36 (1995) , 2761-2776.  doi: 10.1063/1.531064.
      Y. Kurylev  and  M. Lassas , Gelf'and inverse problem for a quadratic operator pencil, Journal of Functional Analysis, 176 (2000) , 247-263.  doi: 10.1006/jfan.2000.3615.
      Y. Kurylev, L. Oksanen and G. Paternain, Inverse problems for the connection Laplacian, to appear in J. Differential Geom., arXiv: 1509.02645.
      I. Lasiecka , J.-L. Lions  and  R. Triggiani , Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986) , 149-192. 
      M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19pp. doi: 10.1088/0266-5611/26/8/085012.
      M. Lassas  and  L. Oksanen , Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., 163 (2014) , 1071-1103. 
      N. Levinson , The inverse Strum-Liouville problem, Mat. Tidsskr. B, (1949) , 25-30. 
      J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅰ, Dunod, Paris, 1968.
      J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅱ, Dunod, Paris, 1968.
      A. Nachman , J. Sylvester  and  G. Uhlmann , An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988) , 595-605.  doi: 10.1007/BF01224129.
      L. Päivärinta  and  V. Serov , An n-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002) , 509-520. 
      L. Robbiano  and  C. Zuily , Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998) , 493-539.  doi: 10.1007/s002220050212.
      W. Rudin, Real and Complex Analysis, McGraw Hill international editions, 1987.
      J. Sylvester  and  G. Uhlmann , A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987) , 153-169.  doi: 10.2307/1971291.
      D. Tataru , Unique continuation for solutions to PDE; between Hörmander's theorem and Holmgren's theorem, Commun. Partial Diff. Eqns., 20 (1995) , 855-884.  doi: 10.1080/03605309508821117.
  • 加载中



Article Metrics

HTML views(1764) PDF downloads(204) Cited by(0)

Access History

Other Articles By Authors

  • on this site
  • on Google Scholar



    DownLoad:  Full-Size Img  PowerPoint