June  2019, 9(2): 289-312. doi: 10.3934/mcrf.2019015

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

1. 

Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

2. 

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

3. 

Department of Mathematics, University College London, London, UK

* Corresponding author

Received  April 2017 Revised  January 2018 Published  November 2018

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.

Citation: Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control and Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015
References:
[1]

C. BardosG. 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.

[2]

M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. 

[3]

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.

[4]

G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1-96.  doi: 10.1007/BF02421600.

[5]

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.

[6]

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. 

[7]

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. 

[8]

L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976.

[9]

H. Isozaki, Some remarks on the multi-dimensional Borg-Levinson theorem, J. Math. Kyoto Univ., 31 (1991), 743-753.  doi: 10.1215/kjm/1250519727.

[10]

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.

[11]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[12]

A. KatchalovY. KurylevM. 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.

[13]

O. KavianY. 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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

Y. Kurylev, L. Oksanen and G. Paternain, Inverse problems for the connection Laplacian, to appear in J. Differential Geom., arXiv: 1509.02645.

[19]

I. LasieckaJ.-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. 

[20]

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.

[21]

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. 

[22]

N. Levinson, The inverse Strum-Liouville problem, Mat. Tidsskr. B, (1949), 25-30. 

[23]

J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅰ, Dunod, Paris, 1968.

[24]

J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅱ, Dunod, Paris, 1968.

[25]

A. NachmanJ. Sylvester and G. Uhlmann, An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.  doi: 10.1007/BF01224129.

[26]

L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520. 

[27]

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.

[28]

W. Rudin, Real and Complex Analysis, McGraw Hill international editions, 1987.

[29]

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.

[30]

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.

show all references

References:
[1]

C. BardosG. 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.

[2]

M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. 

[3]

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.

[4]

G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1-96.  doi: 10.1007/BF02421600.

[5]

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.

[6]

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. 

[7]

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. 

[8]

L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976.

[9]

H. Isozaki, Some remarks on the multi-dimensional Borg-Levinson theorem, J. Math. Kyoto Univ., 31 (1991), 743-753.  doi: 10.1215/kjm/1250519727.

[10]

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.

[11]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[12]

A. KatchalovY. KurylevM. 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.

[13]

O. KavianY. 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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

Y. Kurylev, L. Oksanen and G. Paternain, Inverse problems for the connection Laplacian, to appear in J. Differential Geom., arXiv: 1509.02645.

[19]

I. LasieckaJ.-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. 

[20]

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.

[21]

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. 

[22]

N. Levinson, The inverse Strum-Liouville problem, Mat. Tidsskr. B, (1949), 25-30. 

[23]

J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅰ, Dunod, Paris, 1968.

[24]

J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅱ, Dunod, Paris, 1968.

[25]

A. NachmanJ. Sylvester and G. Uhlmann, An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.  doi: 10.1007/BF01224129.

[26]

L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520. 

[27]

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.

[28]

W. Rudin, Real and Complex Analysis, McGraw Hill international editions, 1987.

[29]

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.

[30]

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.

Figure 1.  Geometric condition (3.10)
Figure 2.  Sets $A_{x, \varepsilon}$
Figure 3.  Support of the geometric optics solution
[1]

Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems and Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449

[2]

Zhiyuan Li, Yikan Liu, Masahiro Yamamoto. Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022027

[3]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[4]

Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems and Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063

[5]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469

[6]

Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021060

[7]

Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems and Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029

[8]

Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems and Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297

[9]

Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems and Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95

[10]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems and Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1

[11]

Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058

[12]

Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems and Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034

[13]

Boya Liu. Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies. Inverse Problems and Imaging, 2020, 14 (5) : 783-796. doi: 10.3934/ipi.2020036

[14]

Giovanni Covi, Keijo Mönkkönen, Jesse Railo. Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems and Imaging, 2021, 15 (4) : 641-681. doi: 10.3934/ipi.2021009

[15]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems and Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229

[16]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems and Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959

[17]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems and Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

[18]

Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems and Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371

[19]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems and Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139

[20]

Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (189)
  • HTML views (627)
  • Cited by (3)

Other articles
by authors

[Back to Top]