On the Inverse Spectral Problem for Graphs with Cycles Pavel Kurasov Lund University, SWEDEN July 17, 2008 Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 1 / 20
Introduction Introduction 1 Quantum graphs Spectral properties Inverse problems The main idea Marchenko-Ostrovsky theory 2 Inverse problems for simple graphs 3 Ring Lassoo Zweih¨ ander General result 4 Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 2 / 20
Introduction Quantum graphs Quantum graph as a triplet Metric graph Γ - union of intervals ∆ j = [ x 2 j − 1 , x 2 j ] connected together at 1 the vertices V m considered as equivalence classes of end-points ⇒ the Hilbert space L 2 (Γ); Differential expression (formally symmetric) on the edges 2 � 2 � − 1 d L q , a = dx + a ( x ) + q ( x ) i ⇒ the linear operator L q , a ; Boundary conditions at the vertices 3 to determine L q , a as a self-adjoint operator, connect together different edges. In this talk we are going to speak only about the standard boundary conditions only, that is: the function is continuous, the sum of ”normal” derivatives is zero Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 3 / 20
Introduction Quantum graphs Quantum graph as a triplet Metric graph Γ - union of intervals ∆ j = [ x 2 j − 1 , x 2 j ] connected together at 1 the vertices V m considered as equivalence classes of end-points ⇒ the Hilbert space L 2 (Γ); Differential expression (formally symmetric) on the edges 2 � 2 � − 1 d L q , a = dx + a ( x ) + q ( x ) i ⇒ the linear operator L q , a ; Boundary conditions at the vertices 3 to determine L q , a as a self-adjoint operator, connect together different edges. In this talk we are going to speak only about the standard boundary conditions only, that is: the function is continuous, the sum of ”normal” derivatives is zero Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 3 / 20
Introduction Spectral properties ”Elimination” of the magnetic field Consider the unitary transformation: � x � � ( U ψ )( x ) = exp − i a ( y ) dy ψ ( x ) , x ∈ ( x 2 n − 1 , x 2 n ) , n = 1 , 2 , ..., N , x 2 n − 1 which allows one to eliminate the magnetic field U − 1 ψ ( x ) = − d 2 � ( − 1 d � dx + a ( x )) 2 + q ( x ) U dx 2 ψ ( x ) + q ( x ) ψ ( x ) . i NB! The magnetic field can be eliminated from the differential expression, but then it appears in the boundary conditions (if the graph is not a tree). Proposition 1. The spectrum of the magnetic Schr¨ odinger operator L q , a is pure discrete and does not depend on the particular form of the magnetic field but just on the fluxes of the magnetic field through the cycles � Φ j = a ( y ) dy . c j Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 4 / 20
Introduction Inverse problems Inverse problems: concise historical overview Solution of the inverse problem for quantum graphs means reconstruction of the metric graph; the differential expressions on the edges; the coupling conditions at the vertices. Obtained results NB! for zero magnetic potential! Reconstruction of the graph: with rationally independent lengths: B. Gutkin, T. Kottos and U. Smilansky, ’99, ’01; P. K., F. Stenberg and M. Nowaczyk ’02, ’05, ’07, ’08; in the case of tree: V. Yurko ’06, S. Avdonin and P. K. ’08; calculation of the Euler characteristic: P. K. ’08, ’08; Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 5 / 20
Introduction Inverse problems Inverse problems: concise historical overview Solution of the inverse problem for quantum graphs means reconstruction of the metric graph; the differential expressions on the edges; the coupling conditions at the vertices. Obtained results NB! for zero magnetic potential! Reconstruction of the graph: with rationally independent lengths: B. Gutkin, T. Kottos and U. Smilansky, ’99, ’01; P. K., F. Stenberg and M. Nowaczyk ’02, ’05, ’07, ’08; in the case of tree: V. Yurko ’06, S. Avdonin and P. K. ’08; calculation of the Euler characteristic: P. K. ’08, ’08; Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 5 / 20
Introduction Inverse problems Reconstruction of the potential on graphs: star graph: N.I. Gerasimenko and B.S. Pavlov, 1988; tree: M. Belishev and A. Vakulenko, ’04, ’06, ’07; M. Brown and R. Weikard, ’05; V. Yurko ’05, ’06, ’08; S. Avdonin and P. K. ’08; impossibility for loops: J. Boman and P. K., 05 V. Pivovarchik, manuscript; Reconstruction of the boundary conditions: for star graphs: V. Kostrykin and R. Schrader ’00, ’06; M. Harmer ’03. Other references: Inverse problems for directed graphs: R. Carlson, ’99. General overview: P. Kuchment ’04. Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 6 / 20
Introduction The main idea The main idea Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field. Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20
Introduction The main idea The main idea Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field. Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20
Introduction The main idea The main idea Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field. Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20
Introduction The main idea The main idea Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field. Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20
Introduction The main idea The main idea Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field. Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20
Marchenko-Ostrovsky theory Marchenko-Ostrovsky theory Provides necessary and sufficient conditions for a sequence of intervals to be the odinger operator L per spectrum of one-dimensional periodic Schr¨ q . Transfer matrix T ( a , b ; λ ) � ψ ( a ) � ψ ( b ) − d 2 � � dx 2 ψ ( x ) + q ( x ) ψ ( x ) = λψ ( x ) ⇒ T ( a , b ; λ ) : �→ ψ ′ ( a ) ψ ′ ( b ) Introduce the functions: u ± ( λ ) = ( t 11 ( λ ) ± t 22 ( λ )) / 2 The end points of the spectral intervals µ j , ˜ µ j are solutions to the equation u + ( λ ) = ± 1 . Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 8 / 20
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