Inverse problems for differential operators on trees with general matching conditions

References

• Montrol , E . 1970 . Quantum theory on a network. . Journal of Mathematical Physics , 11 ( 2 ) : 635 – 648 .
   Web of Science ®Google Scholar
• Faddeev , MD and Pavlov , BS . 1983 . Model of free electrons and the scattering problem. . Teoreticheskaya i Matematicheskaya Fizika , 55 ( 2 ) : 257 – 269 . (Russian); English transl. in Theoretical and Mathematical Physics 55 485–492
   Google Scholar
• Roth , J-P . 1983 . Spectre du laplacien sur un graphe. . Comptes Rendus Mathématique. Académie des Sciences , 296 : 783 – 795 .
   Google Scholar
• Nicaise , S . 1985 . “ Some results on spectral theory over networks, applied to nerve impulse transmission. ” . In Lecture Notes in Mathematics , Vol. 1771 , 532 – 541 . Berlin : Springer .
   Google Scholar
• Penkin , OM and Pokorny , YV . 1988 . On a boundary value problem on a graph. . Differentsial’nye Uravneniya , 24 ( 4 ) : 701 – 703 . (Russian)
   Google Scholar
• von Below , J . 1988 . Sturm–Liouville eigenvalue problem on networks. . Mathematical Methods in the Applied Sciences , 10 : 383 – 395 .
   Google Scholar
• Langese , JE , Leugering , G and Schmidt , JPG . 1994 . Modelling, Analysis and Control of Dynamic Elastic Multi-link Structures , Boston : Birkhäuser .
   Google Scholar
• Ali-Mehmeti , F . 1994 . Nonlinear waves in networks. . Mathematical Research , 80 : 121 – 135 .
   Google Scholar
• Exner , P . 1996 . Contact interactions on graph superlattices. . Journal of Physics A: Mathematical and General , 29 : 87 – 102 .
   Google Scholar
• Kottos , T and Smilansky , U . 1997 . Quantum chaos on graphs. . Physical Review Letters , 79 : 4794 – 4797 .
   Web of Science ®Google Scholar
• Tautz , J , Lindauer , M and Sandeman , DC . 1999 . Transmission of vibration across honeycombs and its detection by bee leg receptors. . Journal of Experimental Biology , 199 : 2585 – 2594 .
   Google Scholar
• Dekoninck , B and Nicaise , S . 2000 . The eigenvalue problem for networks of beams. . Linear Algebra and its Applications , 314 ( 1–3 ) : 165 – 189 .
   Web of Science ®Google Scholar
• Desbois , J . 2000 . Spectral determinant of Schrödinger operators on graphs. . Journal of Physics A: Mathematical and General , 33 : L63 – 67 .
   Google Scholar
• Berkolaiko , G , Bogomolny , E and Keating , J . 2001 . Star graphs and Seba billiards. . Journal of Physics A: Mathematical and General , 34 : 335 – 350 .
   Google Scholar
• Sobolev , A and Solomyak , M . 2002 . Schrödinger operator on homogeneous metric trees: spectrum in gaps. . Reviews in Mathematical Physics , 14 ( 5 ) : 421 – 467 .
   Web of Science ®Google Scholar
• Pokornyi , YV and Borovskikh , AV . 2004 . Differential equations on networks (geometric graphs). . Journal of Mathematical Sciences , 119 ( 6 ) : 691 – 718 .
   Google Scholar
• Pokornyi , Y and Pryadiev , V . 2004 . The qualitative Sturm–Liouville theory on spatial networks. . Journal of Mathematical Sciences , 119 ( 6 ) : 788 – 835 .
   Google Scholar
• Gerasimenko , NI . 1988 . Inverse scattering problem on a noncompact graph. . Teoreticheskaya i Matematicheskaya Fizika , 74 ( 2 ) : 187 – 200 . (Russian); English transl. in Theoretical and Mathematical Physics 75 460–470
   Google Scholar
• Carlson , R . 1999 . Inverse eigenvalue problems on directed graphs. . Transactions of the American Mathematical Society , 351 ( 10 ) : 4069 – 4088 .
   Web of Science ®Google Scholar
• Pivovarchik , VN . 2000 . Inverse problem for the Sturm–Liouville equation on a simple graph. . SIAM Journal on Mathematical Analysis , 32 ( 4 ) : 801 – 819 .
   Web of Science ®Google Scholar
• Kurasov , P and Stenberg , F . 2002 . On the inverse scattering problem on branching graphs. . Journal of Physics A: Mathematical and General , 35 : 101 – 121 .
   Google Scholar
• Belishev , MI . 2004 . Boundary spectral inverse problem on a class of graphs (trees) by the BC method. . Inverse Problems , 20 : 647 – 672 .
   Web of Science ®Google Scholar
• Marchenko , VA . 1986 . Sturm–Liouville Operators and their ApplicationsNaukova Dumka 1977; English transl. , Kiev, Berlin : Birkhäuser .
   Google Scholar
• Levitan , BM . 1987 . Inverse Sturm–Liouville Problems , Moscow : Nauka . 1984; English transl. (Utrecht: VNU Sci. Press) 1987
   Google Scholar
• Pöschel , J and Trubowitz , E . 1987 . Inverse Spectral Theory , New York : Academic Press .
   Google Scholar
• McLaughlin , JR . 1986 . Analytical methods for recovering coefficients in differential equations from spectral data. . SIAM Review , 28 : 53 – 72 .
   Web of Science ®Google Scholar
• Freiling , G and Yurko , VA . 2001 . Inverse Sturm–Liouville Problems and their Applications , New York : NOVA Science Publishers .
   Google Scholar
• Gesztesy , F and Simon , B . 1996 . Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators. . Transactions of the American Mathematical Society , 348 ( 1 ) : 349 – 373 .
   Web of Science ®Google Scholar
• Chadan , K , Colton , D , Paivarinta , L and Rundell , W . 1997 . “ An introduction to inverse scattering and inverse spectral problems. ” . In SIAM Monographs on Mathematical Modelling and Computation , Philadelphia, PA : SIAM .
   Google Scholar
• Rundell , W and Sacks , PE . 1992 . The reconstruction of Sturm–Liouville operators. . Inverse Problems , 8 ( 3 ) : 457 – 482 .
   Web of Science ®Google Scholar
• Yamamoto , M . 1990 . Inverse eigenvalue problem for a vibration of a string with viscous drag. . Journal of Mathematical Analysis and Applications , 152 ( 1 ) : 20 – 34 .
   Web of Science ®Google Scholar
• Hryniv , RO and Mykytyuk , Ya.V. 2003 . Inverse spectral problems for Sturm–Liouville operators with singular potentials. . Inverse Problems , 19 : 665 – 684 .
   Web of Science ®Google Scholar
• Levitan , BM and Sargsyan , IS . 1975 . “ Introduction to Spectral Theory ” . In AMS Translation of Mathematical Monogrphs , Vol. 39 , Providence : American Mathematical Society .
   Google Scholar
• Yurko , VA . 2002 . Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series , Utrecht : VSP .
   Google Scholar
• Yurko , VA . 2005 . Inverse spectral problems for Sturm–Liouville operators on graphs. . Inverse Problems , 21 : 1075 – 1086 .
   Web of Science ®Google Scholar
• Naimark , MA . 1967/1968 . Linear Differential Operators, , 2nd Edn , 1969 Moscow : Nauka . English transl. of 1st Edn, Parts I, II (New York: Ungar)
   Google Scholar
• Conway , JB . 1995 . Functions of One Complex Variable, , 2nd Edn , Vol. I , New York : Springer-Verlag .
   Google Scholar
• Brown , BM and Weikard , R . 2005 . A Borg--Levinson theorem for trees. . Proceedings of the Royal Society of London. Series A Mathematical, Physical and Engineering Sciences , 461 ( 2062 ) : 3231 – 3243 .
   Web of Science ®Google Scholar