zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Reconstruction of a radially symmetric potential from two spectral sequences. (English) Zbl 1011.35129
The authors want to recover the radial potential $q$ in the eigenvalue problem $$ -\Delta u(x)+q(|x|)u(x)=\lambda u(x),\quad \text{ if } |x|<1,\qquad u(x)=0,\quad \text{ if } |x|=1, $$ starting from solutions of the form $u(r,\theta,\varphi)=r^{-1}\psi(r)Y^m_l(\theta,\varphi)$, where $(r,\theta,\varphi)$ are spherical coordinates and $Y^m_l$ are spherical harmonics. \noindent Of course, the family of eigenvalue problems for $\psi$ $$\aligned & A_l(q)u(r)\equiv \psi''(r) + \big[ \lambda - q(r) - l(l+1)r^{-2}\big]\psi(r)=0,\ r\in (0,1),\\ & \psi(1)=0,\ \psi(r)=O(r)\text{ as } r\to 0+,\endaligned\tag 1$$ where $l\in {\bbfN}$, is highly overdetermined. Consequently, the authors consider the problem of recovering $q$ in (1) when a family of spectral data $\{\lambda_{l,n}\}_{(l,n)\in \Lambda}$ is given. Moreover, they introduce the nonlinear operator $$ F_\Lambda(q) = \{ \Psi_1(1,\lambda_{l,n}(q),q)\}_{(l,n)\in \Lambda}, \tag 2$$ $\Psi_1(r,\lambda,q)$ denoting the solution to (1) satisfying the normalization condition $$\lim_{r\to 0+} r^{-l-1}\Psi_1(r,\lambda,q)=1.$$ The main purpose of the paper consists of showing that the linearization of problem (2) around $q=0$ {\it uniquely} determines a small potential $q$. As a consequence, the authors must show that the kernel of the linearized operator, given by $$ D_qF_\Lambda(0)\zeta = \Big\{ c_{l,n}\int_0^1 r^2 j_l^2(\lambda_{l,n}(0)^{1/2}r)\zeta(r) dr\Big\}_{(l,n)\in \Lambda},\qquad c_{l,n}\neq 0,$$ coincides with $\{0\}$, where $j_l$ are the spherical Bessel functions with standard normalization. Moreover, from the asymptotic relationships for the $\lambda_{l,n}(q)$’s it follows $\int_0^1 \zeta(r) dr=0$. \noindent Assuming $\Lambda=\Lambda_{l_1,l_2}=\{l_1,l_2\}\times {\bbfN}$, the assertion will be implied by the (possible) completeness in $H=\big\{\zeta \in L^2(0,1):\int_0^1 \zeta(r) dr=0\big\}$ of the set $\Phi_{l_1,l_2}=\{\varphi_{l}(\lambda_{l,n}(0)^{1/2}r)\}_{(l,n)\in \Lambda_{l_1,l_2}}$, where $\varphi_{l}(r)=rj_l(r)$. However, the authors limit themselves to showing the completeness of the previous set when $\lambda_{l,n}(0)$ is replaced with $(n+l/2)^2\pi^2$, the leading term in its asymptotic expansion. Finally, the authors show that $\Phi_{l,l+1}$ is complete in $H$ if $l=0,1,2,3$. Numerical examples are also provided.

35R30Inverse problems for PDE
31B20Boundary value and inverse problems (higher-dimensional potential theory)
35P05General topics in linear spectral theory of PDE
Full Text: DOI
[1] Abramowitz, M.; Stegun, I.: Handbook of mathematical functions. (1972) · Zbl 0543.33001
[2] Bailey, P. B.; Everitt, W. N.; Zettl, A.: Computing eigenvalues of singular Sturm-Liouville problems. Results math 20, 391-423 (1991) · Zbl 0755.65082
[3] Borg, G.: Eine umkehrung der Sturm--liouvilleschen Êigenwertaufgabe. Acta math 78, 1-96 (1946) · Zbl 0063.00523
[4] Carlson, R.: A borg--Levinson theorem for Bessel operators. Pacific J. Math 177, 1-26 (1997) · Zbl 0868.34061
[5] Carlson, R.; Shubin, C.: Spectral rigidity for radial Schrödinger operators. J. differential equations 113, 338-354 (1994) · Zbl 0809.34088
[6] Gough, D.: Comments on helioseismic inference. Lecture notes in physics 367 (1990)
[7] Global Oscillation Network Group, www.gong.noao.edu
[8] Guillot, J. C.; Ralston, J.: Inverse spectral theory for a singular Sturm--Liouville operator on [0, 1]. J. differential equations 76, 353-373 (1988) · Zbl 0688.34013
[9] Hald, O.: Inverse eigenvalue problem for the mantle. Geophys. J. R. astron. Soc 62, 41-48 (1980) · Zbl 0437.73086
[10] Kirsch, A.: An introduction to the mathematical theory of inverse problems. (1996) · Zbl 0865.35004
[11] Lowe, B.; Pilant, M. S.; Rundell, W.: The recovery of potentials from finite spectral data. SIAM J. Math. anal 23, 482-504 (1992) · Zbl 0763.34005
[12] Nachman, A.; Sylvester, J.; Uhlmann, G.: An n-dimensional borg--Levinson theorem. Comm. math. Phys 115, 595-605 (1988) · Zbl 0644.35095
[13] Pöschel, J.; Trubowitz, E.: Inverse spectral theory. (1987)
[14] Rundell, W.; Sacks, P. E.: Reconstruction techniques for classical inverse Sturm--Liouville problems. Math. comp 58, 161-183 (1992) · Zbl 0745.34015
[15] Young, R.: An introduction to nonharmonic Fourier series. (1980) · Zbl 0493.42001