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M-matrix inverse problem for Sturm-Liouville equations on graphs

Sonja Currie

School of Mathematics University of the Witwatersrand Johannesburg South Africa

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The inverse problem

Consider a Sturm-Liouville boundary value problem on a graph with formally self-adjoint boundary conditions at the nodes. From the M-matrix associated with such a problem we recover, up to a unitary equivalence, the boundary conditions and the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Procedure

Find asymptotics for the M-matrix as the eigenparameter tends to negative infinity. Recover the boundary conditions up to a unitary equivalence from the M-matrix. Show the M-matrix is a Herglotz function. Prove that the poles of the M-matrix are

at the eigenvalues of the associated BVP simple located on the real axis and the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Recover the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

History

The first graph model was used in chemistry - 1930’s by L. Pauling. Development of the theory of differential operators on graphs is recent with most of the research in this area having been done in the last couple of decades. Multi-point boundary value problems and differential systems were studied far earlier then this.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

History

The first graph model was used in chemistry - 1930’s by L. Pauling. Development of the theory of differential operators on graphs is recent with most of the research in this area having been done in the last couple of decades. Multi-point boundary value problems and differential systems were studied far earlier then this.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

History

The first graph model was used in chemistry - 1930’s by L. Pauling. Development of the theory of differential operators on graphs is recent with most of the research in this area having been done in the last couple of decades. Multi-point boundary value problems and differential systems were studied far earlier then this.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Applications

Many applications in physics, engineering and chemistry.

• Eg. scattering theory, quantum wires and quantum chaos,

heat flows in a mesh, photonic crystals. For a survey of the physical systems giving rise to boundary value problems on graphs see P . Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002) R1 -R24 and the bibliography thereof.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Applications

Many applications in physics, engineering and chemistry.

• Eg. scattering theory, quantum wires and quantum chaos,

heat flows in a mesh, photonic crystals. For a survey of the physical systems giving rise to boundary value problems on graphs see P . Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002) R1 -R24 and the bibliography thereof.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Applications

Many applications in physics, engineering and chemistry.

• Eg. scattering theory, quantum wires and quantum chaos,

heat flows in a mesh, photonic crystals. For a survey of the physical systems giving rise to boundary value problems on graphs see P . Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002) R1 -R24 and the bibliography thereof.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Graphs

G denotes a directed graph with a finite number of edges, say K, each of finite length and having the path length metric. Each edge ei of length li can thus be considered as the interval [0, li]. Eg.

✛ r r r ❅ ❅ ❅

• ✠

❅ ❅ ❅ ❘

• r

e2 e1 e3 l1 l3 l4 l2

✛ ✚ ✘ ✙ ✻

e4

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The differential equation

With the above identification we can now consider ly := −d2y dx2 + q(x)y = λy, (1) where q is real valued and essentially bounded

• n the graph G, to be the system of differential equations

− d2yi dx2 + qi(x)yi = λyi (2) for x ∈ [0, li] and i = 1, . . . , K. Here qi and yi are q and y restricted to ei.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Boundary Conditions

Next we specify the boundary conditions at each node ν say. These will depend on the values of y and y′ at ν on each of the incident edges and ultimately we find that the bc’s can be written in the form

K

• j=1

[αijyj + βijy′

j](0) + K

• j=1

[γijyj + δijy′

j](lj) = 0,

(3) i = 1 . . . 2K, 2K = the total no. of linearly independent bc’s. Note that we only consider boundary conditions which are self-adjoint w.r.t. l and L2(G).

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Operator eigenvalue problem

The above BVP on G can be reformulated as an operator eigenvalue problem by setting Lf := −f ′′ + qf with domain D(L) = {f | f, f ′ ∈ AC, L(f) ∈ L2(G), f obeys (3)}.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Separated boundary conditions

The above BVP on G is also equivalent to the formally self-adjoint system −MT′′ + PT = λT (4) with separated boundary conditions A∗T(0) − B∗T′(0) = 0, (5) Γ∗T(1) − ∆∗T′(1) = (6)

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Separated boundary conditions

where M = 4diag

• 1

l2

1 , . . . , 1

l2

k , . . . , 1

l2

1 , . . . , 1

l2

k

• ,

P is a diagonal, 2K × 2K, matrix dependent on the potential on each edge of the graph, A∗ = I −I

• , −B∗ =

I I

• ,and

Γ, ∆ are real, constant 2K × 2K matrices.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

We define the matrix Prüfer angle in the same way as Atkinson, Barret and Etgen. F .V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, (1964) J.H. Barrett, A Prüfer transformation for matrix differential equations, Proc. Amer. Math. Soc., 8 (1957), 510-518 G.J. Etgen, Two point boundary problems for second order matrix differential systems, Transactions of the Amer.

• Math. Soc., 149 (1970), 119-132

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

We define the matrix Prüfer angle in the same way as Atkinson, Barret and Etgen. F .V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, (1964) J.H. Barrett, A Prüfer transformation for matrix differential equations, Proc. Amer. Math. Soc., 8 (1957), 510-518 G.J. Etgen, Two point boundary problems for second order matrix differential systems, Transactions of the Amer.

• Math. Soc., 149 (1970), 119-132

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

We define the matrix Prüfer angle in the same way as Atkinson, Barret and Etgen. F .V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, (1964) J.H. Barrett, A Prüfer transformation for matrix differential equations, Proc. Amer. Math. Soc., 8 (1957), 510-518 G.J. Etgen, Two point boundary problems for second order matrix differential systems, Transactions of the Amer.

• Math. Soc., 149 (1970), 119-132

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

The second order equation (4) can be rewritten as the following first order system Y′ = Z and Z′ = −G(x, λ)Y where G(x, λ) = M−1(λ − P). Without loss of generality we may then assume that the following three properties hold:

G(x, λ) must be continuous and symmetric. A∗B = B∗A and Γ∗∆ = ∆∗Γ. A∗A + B∗B = I and Γ∗Γ + ∆∗∆ = I.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

The second order equation (4) can be rewritten as the following first order system Y′ = Z and Z′ = −G(x, λ)Y where G(x, λ) = M−1(λ − P). Without loss of generality we may then assume that the following three properties hold:

G(x, λ) must be continuous and symmetric. A∗B = B∗A and Γ∗∆ = ∆∗Γ. A∗A + B∗B = I and Γ∗Γ + ∆∗∆ = I.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

The second order equation (4) can be rewritten as the following first order system Y′ = Z and Z′ = −G(x, λ)Y where G(x, λ) = M−1(λ − P). Without loss of generality we may then assume that the following three properties hold:

G(x, λ) must be continuous and symmetric. A∗B = B∗A and Γ∗∆ = ∆∗Γ. A∗A + B∗B = I and Γ∗Γ + ∆∗∆ = I.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

The second order equation (4) can be rewritten as the following first order system Y′ = Z and Z′ = −G(x, λ)Y where G(x, λ) = M−1(λ − P). Without loss of generality we may then assume that the following three properties hold:

G(x, λ) must be continuous and symmetric. A∗B = B∗A and Γ∗∆ = ∆∗Γ. A∗A + B∗B = I and Γ∗Γ + ∆∗∆ = I.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

The second order equation (4) can be rewritten as the following first order system Y′ = Z and Z′ = −G(x, λ)Y where G(x, λ) = M−1(λ − P). Without loss of generality we may then assume that the following three properties hold:

G(x, λ) must be continuous and symmetric. A∗B = B∗A and Γ∗∆ = ∆∗Γ. A∗A + B∗B = I and Γ∗Γ + ∆∗∆ = I.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The matrix Prüfer angle

The matrix Prüfer angle corresponding to the system with separated bc’s, (4)-(6), is then given by F(x, λ) = (V − iU)−1(V + iU) where U(x, λ) = S(x, λ)Γ − C(x, λ)∆, V(x, λ) = C(x, λ)Γ + S(x, λ)∆ and where {S(x, λ), C(x, λ)} is the solution of S′ = H(x, λ)C, C′ = −H(x, λ)S, S(0, λ) = B∗, C(0, λ) = A∗. and H(x, λ) = CC∗ + SGS∗.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

Let W1 be a solution of (4) satisfying, at 0, the boundary conditions W1(0, λ) = B and W′

1(0, λ) = A

that is A∗W1(0, λ) − B∗W′

1(0, λ) = 0,

and let χ(x, λ) = W1(x, λ)[∆∗W′

1(1, λ) − Γ∗W1(1, λ)]−1,

then set R = −Γ ∆ ∆ Γ

• and

W(x, λ) = W2(x, λ) W3(x, λ) W′

2(x, λ)

W′

3(x, λ)

• where W2 and W3 are solutions of (4) obeying the terminal

conditions W(1, λ) = R.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

Let W1 be a solution of (4) satisfying, at 0, the boundary conditions W1(0, λ) = B and W′

1(0, λ) = A

that is A∗W1(0, λ) − B∗W′

1(0, λ) = 0,

and let χ(x, λ) = W1(x, λ)[∆∗W′

1(1, λ) − Γ∗W1(1, λ)]−1,

then set R = −Γ ∆ ∆ Γ

• and

W(x, λ) = W2(x, λ) W3(x, λ) W′

2(x, λ)

W′

3(x, λ)

• where W2 and W3 are solutions of (4) obeying the terminal

conditions W(1, λ) = R.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

Let W1 be a solution of (4) satisfying, at 0, the boundary conditions W1(0, λ) = B and W′

1(0, λ) = A

that is A∗W1(0, λ) − B∗W′

1(0, λ) = 0,

and let χ(x, λ) = W1(x, λ)[∆∗W′

1(1, λ) − Γ∗W1(1, λ)]−1,

then set R = −Γ ∆ ∆ Γ

• and

W(x, λ) = W2(x, λ) W3(x, λ) W′

2(x, λ)

W′

3(x, λ)

• where W2 and W3 are solutions of (4) obeying the terminal

conditions W(1, λ) = R.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

We define the Weyl M-matrix, M(λ), of (4)-(6) to be given by Ψ = W2 + W3M(λ). (7) where Ψ obeys (5).

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

The M-matrix defined in this way exists and is well-defined for λ not an eigenvalue of (4)-(6). This is a generalisation of the m-function for the classical scalar Sturm-Liouville operator and is consistant with the M-matrix for general systems given by Krall and the M-vector for trees given by Yurko in

A.M. Krall, Hilbert Space, Boundary value problems and

• rthogonal polynomials, Birkhäuser Verlag, Basel - Boston
• Berlin, (2002)
• V. Yurko, Inverse spectral problems for Sturm-Liouville
• perators on graphs, Inverse problems, 21 (2005),

1075-1086

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

The M-matrix defined in this way exists and is well-defined for λ not an eigenvalue of (4)-(6). This is a generalisation of the m-function for the classical scalar Sturm-Liouville operator and is consistant with the M-matrix for general systems given by Krall and the M-vector for trees given by Yurko in

A.M. Krall, Hilbert Space, Boundary value problems and

• rthogonal polynomials, Birkhäuser Verlag, Basel - Boston
• Berlin, (2002)
• V. Yurko, Inverse spectral problems for Sturm-Liouville
• perators on graphs, Inverse problems, 21 (2005),

1075-1086

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

The M-matrix defined in this way exists and is well-defined for λ not an eigenvalue of (4)-(6). This is a generalisation of the m-function for the classical scalar Sturm-Liouville operator and is consistant with the M-matrix for general systems given by Krall and the M-vector for trees given by Yurko in

A.M. Krall, Hilbert Space, Boundary value problems and

• rthogonal polynomials, Birkhäuser Verlag, Basel - Boston
• Berlin, (2002)
• V. Yurko, Inverse spectral problems for Sturm-Liouville
• perators on graphs, Inverse problems, 21 (2005),

1075-1086

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

The M-matrix defined in this way exists and is well-defined for λ not an eigenvalue of (4)-(6). This is a generalisation of the m-function for the classical scalar Sturm-Liouville operator and is consistant with the M-matrix for general systems given by Krall and the M-vector for trees given by Yurko in

A.M. Krall, Hilbert Space, Boundary value problems and

• rthogonal polynomials, Birkhäuser Verlag, Basel - Boston
• Berlin, (2002)
• V. Yurko, Inverse spectral problems for Sturm-Liouville
• perators on graphs, Inverse problems, 21 (2005),

1075-1086

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

In the above reference Yurko solves the inverse problem of recovering the operator from the M-vector for Sturm-Liouville operators on trees with continuity and Kirchhoff bc’s at the nodes. Recently Yurko has also considered inverse problems for graphs which may contain a cycle but still with continuity and Kirchhoff bc’s at the nodes, see V. Yurko, Inverse problems for Sturm-Liouville operators on graphs with a cycle, Operators and Matrices, Vol. 2 no. 4 (2008), 543-553 The problem we wish to solve is thus a generalisation of Yurko’s results since we consider general self-adjoint bc’s for the recovery of the bc’s and general co-normal bc’s for the recovery of the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

In the above reference Yurko solves the inverse problem of recovering the operator from the M-vector for Sturm-Liouville operators on trees with continuity and Kirchhoff bc’s at the nodes. Recently Yurko has also considered inverse problems for graphs which may contain a cycle but still with continuity and Kirchhoff bc’s at the nodes, see V. Yurko, Inverse problems for Sturm-Liouville operators on graphs with a cycle, Operators and Matrices, Vol. 2 no. 4 (2008), 543-553 The problem we wish to solve is thus a generalisation of Yurko’s results since we consider general self-adjoint bc’s for the recovery of the bc’s and general co-normal bc’s for the recovery of the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

The M-matrix

In the above reference Yurko solves the inverse problem of recovering the operator from the M-vector for Sturm-Liouville operators on trees with continuity and Kirchhoff bc’s at the nodes. Recently Yurko has also considered inverse problems for graphs which may contain a cycle but still with continuity and Kirchhoff bc’s at the nodes, see V. Yurko, Inverse problems for Sturm-Liouville operators on graphs with a cycle, Operators and Matrices, Vol. 2 no. 4 (2008), 543-553 The problem we wish to solve is thus a generalisation of Yurko’s results since we consider general self-adjoint bc’s for the recovery of the bc’s and general co-normal bc’s for the recovery of the potential.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Relating F(1) and the M-matrix

Theorem The M-matrix satisfies M∗(λ) = i(F∗(1, λ) − I)−1(F∗(1, λ) + I). The poles of the determinant of M∗ are prescisely the eigenvalues of (4)-(6). Corollary The matrix Prüfer angle F(1, λ) determines M(λ) and vice-versa.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Relating F(1) and the M-matrix

Theorem The M-matrix satisfies M∗(λ) = i(F∗(1, λ) − I)−1(F∗(1, λ) + I). The poles of the determinant of M∗ are prescisely the eigenvalues of (4)-(6). Corollary The matrix Prüfer angle F(1, λ) determines M(λ) and vice-versa.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Asymptotics

Theorem Let λ = −σ2 then the matrix Prüfer angle, F, obeys the following asymptotic approximation as σ → ∞ F∗(1, −σ2) = (Γ∗ + i∆∗) H (Γ∗ − i∆∗)−1 , where H = diag

• 1 + O

1

σ

• , . . . , 1 + O

1

σ

• Corollary

Asymptotically as σ → ∞, M∗(λ) takes the form M∗(λ) = i

• 2i∆ + O

1 σ −1 2Γ + O 1 σ

• .

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Asymptotics

Theorem Let λ = −σ2 then the matrix Prüfer angle, F, obeys the following asymptotic approximation as σ → ∞ F∗(1, −σ2) = (Γ∗ + i∆∗) H (Γ∗ − i∆∗)−1 , where H = diag

• 1 + O

1

σ

• , . . . , 1 + O

1

σ

• Corollary

Asymptotically as σ → ∞, M∗(λ) takes the form M∗(λ) = i

• 2i∆ + O

1 σ −1 2Γ + O 1 σ

• .

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Recovery of the boundary conditions

Theorem Let (Γ∗, ∆∗, P) denote the BVP (4)-(6) and (˜ Γ∗, ˜ ∆∗, ˜ P) the BVP (4)-(6) but with Γ replaced by ˜ Γ, ∆ by ˜ ∆ and P by ˜ P. If the problems (Γ∗, ∆∗, P) and (˜ Γ∗, ˜ ∆∗, ˜ P) have the same M-matrix, M(λ), where ˜ Γ∗ ˜ ∆ = ˜ ∆∗˜ Γ and ˜ Γ∗˜ Γ + ˜ ∆∗ ˜ ∆ = I. Then ∆ = U ˜ ∆ and Γ = U˜ Γ where U = Γ˜ Γ∗ + ∆ ˜ ∆∗ is unitary.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Herglotz function

Theorem The M-matrix, M(λ), is a Herglotz function of rank 2K. Clark and Gesztesy, Weyl-Titchmarsh M-function asymptotics for matrix valued Schrodinger operators, Proc. London math. Soc., 3 (2001), No. 06, 701–724.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Wronskian identities

Lemma For λ ∈ R, the following Wronskian identities hold: W′∗

3 (x)W2(x) − W∗ 3(x)W′ 2(x)

= −I, W∗

2(x)W′ 3(x) − W′∗ 2 (x)W3(x)

= −I, W′∗

2 (x)W3(x) − W∗ 2(x)W′ 3(x)

= I, W∗

3(x)W′ 2(x) − W′∗ 3 (x)W2(x)

= I, W′∗

2 (x)W2(x) − W∗ 2(x)W′ 2(x)

= 0, W′∗

3 (x)W3(x) − W∗ 3(x)W′ 3(x)

= 0.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Green’s function

Lemma The Green’s function for the boundary value problem (4)-(6) can be represented as G(x, t) = W3(x)Ψ∗(t)M−1, t < x Ψ(x)W∗

3(t)M−1,

t > x . (8)

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Herglotz representation

As the M-matrix defined in (7) is a Herglotz function it admits the following representation, M(λ) = C + Dλ +

• λn

Mn

• 1

λn − λ − λn 1 + λ2

n

• ,

where C = Re(M(i)) and D = limη→∞( 1

iηM(iη)) ≥ 0. Thus

lim

λ→λn(λ − λn)M(λ) = −Mn,

i.e. −Mn is the residue of the pole of M(λ) at λn.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Nature of the poles

Theorem The poles of the M-matrix are simple, located on the real axis and are the eigenvalues of (4)-(6). The residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Proof

Since M(λ) is a matrix-valued Herglotz function the poles

• f the M-matrix are simple and located on the real axis and

the residue at a pole is a negative semi-definite matrix - F .Gesztesy, E. Tsekanovskii, On matrix valued Herglotz functions,Math. Nachr., 218 (2000), 61-138. By the definition of the M-matrix all the poles of M are eigenvalues of (4)-(6). At an eigenvalue of (4)-(6) the Green’s function has a pole, giving that if λ is an eigenvalue then λ is a pole of G(x, t) and thus, by (8), λ is a pole of Ψ and hence, is a pole of M.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Proof

Since M(λ) is a matrix-valued Herglotz function the poles

• f the M-matrix are simple and located on the real axis and

the residue at a pole is a negative semi-definite matrix - F .Gesztesy, E. Tsekanovskii, On matrix valued Herglotz functions,Math. Nachr., 218 (2000), 61-138. By the definition of the M-matrix all the poles of M are eigenvalues of (4)-(6). At an eigenvalue of (4)-(6) the Green’s function has a pole, giving that if λ is an eigenvalue then λ is a pole of G(x, t) and thus, by (8), λ is a pole of Ψ and hence, is a pole of M.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Proof

If Fn,j, j = 1, . . . , νn, is an orthonormal sequence of eigenfunctions corresponding to λn (λn not repeated according to multiplicity) where νn is the multiplicity of the eigenvalue λn then Fn,j can be written as Fn,j(x) = W3(λn, x)cn,j where cn,j is a column vector. The form which the negative semi-definite matrix −Mn takes is then −

νn

• j=1

cn,jc∗

n,j = −Mn.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Proof

If Fn,j, j = 1, . . . , νn, is an orthonormal sequence of eigenfunctions corresponding to λn (λn not repeated according to multiplicity) where νn is the multiplicity of the eigenvalue λn then Fn,j can be written as Fn,j(x) = W3(λn, x)cn,j where cn,j is a column vector. The form which the negative semi-definite matrix −Mn takes is then −

νn

• j=1

cn,jc∗

n,j = −Mn.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Proof

Define Cn := [cn,1, . . . , cn,νn, 0, . . . , 0]. Then the rank of Cn is νn. Also the rank of Cn is equal to the number of non-zero eigenvalues of Cn, counted by multiplicity. Denote these eigenvalues by µ1, . . . , µνn. Then CnC∗

n has non-zero eigenvalues |µ1|2, . . . , |µνn|2. Thus

CnC∗

n has rank νn and Rank(−Mn) = νn.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Proof

Define Cn := [cn,1, . . . , cn,νn, 0, . . . , 0]. Then the rank of Cn is νn. Also the rank of Cn is equal to the number of non-zero eigenvalues of Cn, counted by multiplicity. Denote these eigenvalues by µ1, . . . , µνn. Then CnC∗

n has non-zero eigenvalues |µ1|2, . . . , |µνn|2. Thus

CnC∗

n has rank νn and Rank(−Mn) = νn.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Co-normal boundary conditions

Definition For the system formulation (4)-(6) the bc’s at x = 1, are co-normal if and only if Γ and ∆ are such that, when S =    u u′

• ∈

C2K ⊕ C2K | Γ∗u − ∆∗u′ = 0    , is such that there exists a subspace N, of dimension n, of C2K so that u

• ∈ S for all u ∈ N and there exists a real diagonal

matrix D =: diag{d1, . . . , d2K} such that u u′

• ∈ S if and only if

u ∈ N and (Du − u′) · v = 0 for all v ∈ N.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Co-normal boundary conditions

Lemma Suppose that the boundary conditions at x = 1, are co-normal and that S, N, D are as given in the definition of co-normal bc’s. Then there exists an orthonormal basis w1, . . . , w2K for C2K and real numbers µ1, . . . , µn such that, without loss of generality, ∆ and Γ may be written as ∆ =

• w1
• 1 + |µ1|2 , . . . ,

wn

• 1 + |µn|2 , 0, . . . , 0
• (9)

and Γ =

• µ1w1
• 1 + |µ1|2 , . . . ,

µnwn

• 1 + |µn|2 , wn+1, . . . , w2K
• .

(10)

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Co-normal boundary conditions

Co-normal boundary conditions on a graph correspond in nature to co-normal (non-oblique) boundary conditions for elliptic partial differential operators. Most physically interesting boundary conditions on graphs fall into the co-normal category. In particular, ‘Kirchhoff’, Dirichlet, Neumann and periodic boundary conditions are all co-normal, but this class does not include all self-adjoint boundary-value problems on graphs.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Co-normal boundary conditions

Co-normal boundary conditions on a graph correspond in nature to co-normal (non-oblique) boundary conditions for elliptic partial differential operators. Most physically interesting boundary conditions on graphs fall into the co-normal category. In particular, ‘Kirchhoff’, Dirichlet, Neumann and periodic boundary conditions are all co-normal, but this class does not include all self-adjoint boundary-value problems on graphs.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Mar˘ cenko’s result

Lemma There exists a kernel, kh,m,q(t, y), (k∞,m,q(t, y) resp.) such that, vh(∞),m,q[f](t) := 1

t

kh(∞),m,q(t, y)f(y) dy, defines a continuous linear transformation on L2[0, 1], and if yλ is the solution of −my′′

λ = λyλ on [0, 1] with y′ λ(1) = hyλ(1)

(yλ(1) = 0 resp.), for m > 0 a real constant, then zλ := (I + vh,m,q)yλ (zλ := (I + v∞,m,q)yλ resp.) is the solution of −mz′′

λ + qzλ = λzλ on [0, 1] with z′ λ(1) = hyλ(1)

and zλ(1) = yλ(1) (zλ(1) = 0 and z′

λ(1) = y′ λ(1) resp.), for each

λ ∈ R.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Unitary operator

Lemma Let the problems (Γ∗, ∆∗, P) and (˜ Γ∗, ˜ ∆∗, ˜ P) have the same M-matrix. If there exists a linear continuous transformation

• perator, H, on L2[0, 1], independent of λ, which maps

H[W3(λ, x)] = ˜ W3(λ, x), (11) where ˜ W3(λ, x) is the solution to (˜ Γ∗, ˜ ∆∗, ˜ P) obeying ˜ W3(λ, 1) = ˜ ∆ and ˜ W3

′(λ, 1) = ˜

Γ, then H is unitary.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Recovery of the operator

Theorem If the problems (Γ∗, ∆∗, P) and (˜ Γ∗, ˜ ∆∗, ˜ P) have the same M-matrix, and if we assume that the boundary conditions at x = 1 are co-normal and that the weight matrix M commutes with Γ, ∆, ˜ Γ, ˜ ∆, then P = U˜ PU∗ .

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Since the bc’s at x = 1 are co-normal and M commutes with Γ, ∆, ˜ Γ, ˜ ∆ we can use Mar˘ cenko’s result and ultimately show that there exits a Volterra map VP,M such that if ¯ Y(t) is the solution of −M¯ Y(t)′′ = λ¯ Y(t) with ¯ Y(1) = ∆ and ¯ Y′(1) = Γ, then ¯ Z(t) := (I + VP,M)¯ Y(t) is the solution of −M¯ Z(t)′′ + P¯ Z(t) = λ¯ Z(t) with ¯ Z(1) = ∆ and ¯ Z′(1) = Γ.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Now (I + VP,M)−1 = I + WP,M, where WP,M is Volterra - Mar˘ cenko So if Y is the solution of −MY′′ + PY = λY on [0, 1] with Y(1) = ∆ and Y′(1) = Γ then Z := (I + WP,M)Y is the solution of −MZ′′ = λZ on [0, 1] with Z(1) = ∆ and Z′(1) = Γ. Giving ˜ Z := U∗(I + WP,M)Y = U∗Z is the solution of −U∗MU(U∗Z)′′ = λ(U∗Z) with ˜ Z(1) = U∗Z(1) = U∗∆ = ˜ ∆ and ˜ Z′(1) = U∗Z′(1) = U∗Γ = ˜ Γ.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Now (I + VP,M)−1 = I + WP,M, where WP,M is Volterra - Mar˘ cenko So if Y is the solution of −MY′′ + PY = λY on [0, 1] with Y(1) = ∆ and Y′(1) = Γ then Z := (I + WP,M)Y is the solution of −MZ′′ = λZ on [0, 1] with Z(1) = ∆ and Z′(1) = Γ. Giving ˜ Z := U∗(I + WP,M)Y = U∗Z is the solution of −U∗MU(U∗Z)′′ = λ(U∗Z) with ˜ Z(1) = U∗Z(1) = U∗∆ = ˜ ∆ and ˜ Z′(1) = U∗Z′(1) = U∗Γ = ˜ Γ.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Now (I + VP,M)−1 = I + WP,M, where WP,M is Volterra - Mar˘ cenko So if Y is the solution of −MY′′ + PY = λY on [0, 1] with Y(1) = ∆ and Y′(1) = Γ then Z := (I + WP,M)Y is the solution of −MZ′′ = λZ on [0, 1] with Z(1) = ∆ and Z′(1) = Γ. Giving ˜ Z := U∗(I + WP,M)Y = U∗Z is the solution of −U∗MU(U∗Z)′′ = λ(U∗Z) with ˜ Z(1) = U∗Z(1) = U∗∆ = ˜ ∆ and ˜ Z′(1) = U∗Z′(1) = U∗Γ = ˜ Γ.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Since UM = MU we get that ˜ Z is the solution of −M˜ Z′′ = λ˜ Z with ˜ Z(1) = ˜ ∆ and ˜ Z′(1) = ˜ Γ. Let ˜ Y := (I + V˜

P,M)˜

Z = (I + V˜

P,M)U∗(I + WP,M)Y,

then ˜ Y is the solution of −M˜ Y′′ + ˜ P˜ Y = λ˜ Y with ˜ Y(1) = ˜ ∆ and ˜ Y′(1) = ˜ Γ. Let HY := (I + V˜

P,M)U∗(I + WP,M)Y for Y ∈ L2[0, 1]. If Y is

any solution of −MY′′ + PY = λY, then ˜ Y := HY is the solution of −M˜ Y′′ + ˜ P˜ Y = λ˜ Y with ˜ Y(1) = U∗Y(1) and ˜ Y′(1) = U∗Y′(1).

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Since UM = MU we get that ˜ Z is the solution of −M˜ Z′′ = λ˜ Z with ˜ Z(1) = ˜ ∆ and ˜ Z′(1) = ˜ Γ. Let ˜ Y := (I + V˜

P,M)˜

Z = (I + V˜

P,M)U∗(I + WP,M)Y,

then ˜ Y is the solution of −M˜ Y′′ + ˜ P˜ Y = λ˜ Y with ˜ Y(1) = ˜ ∆ and ˜ Y′(1) = ˜ Γ. Let HY := (I + V˜

P,M)U∗(I + WP,M)Y for Y ∈ L2[0, 1]. If Y is

any solution of −MY′′ + PY = λY, then ˜ Y := HY is the solution of −M˜ Y′′ + ˜ P˜ Y = λ˜ Y with ˜ Y(1) = U∗Y(1) and ˜ Y′(1) = U∗Y′(1).

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Since UM = MU we get that ˜ Z is the solution of −M˜ Z′′ = λ˜ Z with ˜ Z(1) = ˜ ∆ and ˜ Z′(1) = ˜ Γ. Let ˜ Y := (I + V˜

P,M)˜

Z = (I + V˜

P,M)U∗(I + WP,M)Y,

then ˜ Y is the solution of −M˜ Y′′ + ˜ P˜ Y = λ˜ Y with ˜ Y(1) = ˜ ∆ and ˜ Y′(1) = ˜ Γ. Let HY := (I + V˜

P,M)U∗(I + WP,M)Y for Y ∈ L2[0, 1]. If Y is

any solution of −MY′′ + PY = λY, then ˜ Y := HY is the solution of −M˜ Y′′ + ˜ P˜ Y = λ˜ Y with ˜ Y(1) = U∗Y(1) and ˜ Y′(1) = U∗Y′(1).

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

In particular HW3 = ˜ W3 and HW2 = ˜ W2, and hence H is unitary. So (it can be shown that) U∗ = H = (I + V˜

P,M)U∗(I + WP,M).

Therefore ˜ W3 = U∗W3 and ˜ W2 = U∗W2, and − M(U∗Wj)′′ + ˜ PU∗Wj = λU∗Wj, (12) for j = 2, 3.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

In particular HW3 = ˜ W3 and HW2 = ˜ W2, and hence H is unitary. So (it can be shown that) U∗ = H = (I + V˜

P,M)U∗(I + WP,M).

Therefore ˜ W3 = U∗W3 and ˜ W2 = U∗W2, and − M(U∗Wj)′′ + ˜ PU∗Wj = λU∗Wj, (12) for j = 2, 3.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

In particular HW3 = ˜ W3 and HW2 = ˜ W2, and hence H is unitary. So (it can be shown that) U∗ = H = (I + V˜

P,M)U∗(I + WP,M).

Therefore ˜ W3 = U∗W3 and ˜ W2 = U∗W2, and − M(U∗Wj)′′ + ˜ PU∗Wj = λU∗Wj, (12) for j = 2, 3.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Premultiplying equation (12) by U and noting that MU = UM, gives − MW′′

j + U˜

PU∗Wj = λWj, (13) for j = 2, 3. We also have that − MW′′

j + PWj = λWj,

(14) for j = 2, 3. So (U˜ PU∗ − P)Wj = 0, for j = 2, 3. Since the column space of [W2, W3] spans the solution space of −MY′′ + PY = λY, we can show that P = U˜ PU∗.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Premultiplying equation (12) by U and noting that MU = UM, gives − MW′′

j + U˜

PU∗Wj = λWj, (13) for j = 2, 3. We also have that − MW′′

j + PWj = λWj,

(14) for j = 2, 3. So (U˜ PU∗ − P)Wj = 0, for j = 2, 3. Since the column space of [W2, W3] spans the solution space of −MY′′ + PY = λY, we can show that P = U˜ PU∗.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Premultiplying equation (12) by U and noting that MU = UM, gives − MW′′

j + U˜

PU∗Wj = λWj, (13) for j = 2, 3. We also have that − MW′′

j + PWj = λWj,

(14) for j = 2, 3. So (U˜ PU∗ − P)Wj = 0, for j = 2, 3. Since the column space of [W2, W3] spans the solution space of −MY′′ + PY = λY, we can show that P = U˜ PU∗.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Outline of proof

Premultiplying equation (12) by U and noting that MU = UM, gives − MW′′

j + U˜

PU∗Wj = λWj, (13) for j = 2, 3. We also have that − MW′′

j + PWj = λWj,

(14) for j = 2, 3. So (U˜ PU∗ − P)Wj = 0, for j = 2, 3. Since the column space of [W2, W3] spans the solution space of −MY′′ + PY = λY, we can show that P = U˜ PU∗.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Note

For the above result, we only require Mn = ˜ Mn and not that the entire M-matrices are equal.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Recovery of the operator

Corollary If all the edges of the graph G have the same length, l, and the systems problems (Γ∗, ∆∗, P) and (˜ Γ∗, ˜ ∆∗, ˜ P) have the same M-matrix, then ∆ = U ˜ ∆, Γ = U˜ Γ and P = U˜ PU∗ where U = Γ˜ Γ∗ + ∆ ˜ ∆∗ is a unitary matrix. Corollary If the problems (Γ∗, ∆∗, P) and (Γ∗, ∆∗, ˜ P) have the same M-matrix and M commutes with Γ, ∆ then P = ˜ P.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Recovery of the operator

Corollary If all the edges of the graph G have the same length, l, and the systems problems (Γ∗, ∆∗, P) and (˜ Γ∗, ˜ ∆∗, ˜ P) have the same M-matrix, then ∆ = U ˜ ∆, Γ = U˜ Γ and P = U˜ PU∗ where U = Γ˜ Γ∗ + ∆ ˜ ∆∗ is a unitary matrix. Corollary If the problems (Γ∗, ∆∗, P) and (Γ∗, ∆∗, ˜ P) have the same M-matrix and M commutes with Γ, ∆ then P = ˜ P.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Conclusion

Note that given a set of eigenvalues λn and the terminal value, ∆∗Fn,j(1) + Γ∗F′

n,j(1) = cn,j,

Mn is uniquely determined and the above corollaries apply. The above note is actually a more appealing result since it means that from the eigenvalues and the data at the nodes

• f the given graph, i.e. the terminal conditions, we can

recover the boundary conditions and the potential. It does not rely on the superficial nodes inserted into each edge

• nly on the original given nodes.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs

Conclusion

Note that given a set of eigenvalues λn and the terminal value, ∆∗Fn,j(1) + Γ∗F′

n,j(1) = cn,j,

Mn is uniquely determined and the above corollaries apply. The above note is actually a more appealing result since it means that from the eigenvalues and the data at the nodes

• f the given graph, i.e. the terminal conditions, we can

recover the boundary conditions and the potential. It does not rely on the superficial nodes inserted into each edge

• nly on the original given nodes.

Sonja Currie M-matrix inverse problem for Sturm-Liouville equations on graphs