Stability of Eigenvalues of Quantum Graphs with Respect to Magnetic - PowerPoint PPT Presentation

Stability of Eigenvalues of Quantum Graphs with Respect to Magnetic Perturbation Tracy Weyand Texas A&M University College Station, TX 77843-3368 www.math.tamu.edu/ ˜ tweyand tweyand@math.tamu.edu arXiv:1212.4475 , Phil Trans Roy Soc A (joint with G. Berkolaiko) Texas Analysis and Mathematical Physics Symposium, 2013

Metric Graphs Γ = { V , E , L } Compact Weyand Eigenvalues of Quantum Graphs

Metric Graphs Γ = { V , E , L } Compact Functions: � H 2 (Γ) = ⊕ e ∈ E H 2 ( e ) 1 st Betti # = | E | − | V | + 1 Weyand Eigenvalues of Quantum Graphs

Quantum Graphs Metric Graph + Differential Operator Schr¨ odinger Operator H 0 (Γ) : f �→ − d 2 f ∈ � H 2 (Γ , C ) dx 2 f ( x ) + q ( x ) f ( x ) , � f ( x ) is continuous at v , � � � d dx e f ( x ) � v = χ v f ( v ) , χ v ∈ R e ∈ E v Weyand Eigenvalues of Quantum Graphs

Quantum Graphs Metric Graph + Differential Operator Schr¨ odinger Operator H 0 (Γ) : f �→ − d 2 f ∈ � H 2 (Γ , C ) dx 2 f ( x ) + q ( x ) f ( x ) , � f ( x ) is continuous at v , � � � d dx e f ( x ) � v = χ v f ( v ) , χ v ∈ R e ∈ E v Magnetic Schr¨ odinger Operator � d � 2 f ∈ � H 2 (Γ , C ) H A (Γ) : f �→ − dx − iA ( x ) f ( x ) + q ( x ) f ( x ) , � f ( x ) is continuous at v , � � � � � d dx e − iA e ( x ) f ( x ) v = χ v f ( v ) , χ v ∈ R � e ∈ E v Weyand Eigenvalues of Quantum Graphs

Magnetic Flux c 1 + c 2 c 1 + c c 2 c 1 - c 1 c 2 2 - � c + j α j = A ( x ) dx mod 2 π c − j Magnetic Flux : α = ( α 1 , α 2 , . . . , α β ) Weyand Eigenvalues of Quantum Graphs

Unitarily Equivalent Operators � d � 2 f ∈ � H 2 (Γ , C ) H A (Γ) : f �→ − dx − iA ( x ) f ( x ) + q ( x ) f ( x ) , � f ( x ) is continuous at v , � � � � � d dx e − iA e ( x ) f ( x ) v = χ v f ( v ) , χ v ∈ R � e ∈ E v Weyand Eigenvalues of Quantum Graphs

Unitarily Equivalent Operators � d � 2 f ∈ � H 2 (Γ , C ) H A (Γ) : f �→ − dx − iA ( x ) f ( x ) + q ( x ) f ( x ) , � f ( x ) is continuous at v , � � � � � d dx e − iA e ( x ) f ( x ) v = χ v f ( v ) , χ v ∈ R � e ∈ E v H α (Γ) : f �→ − d 2 f ∈ � H 2 ( T , C ) dx 2 f ( x ) + q ( x ) f ( x ) ,  f ( x ) is continuous at v   �  df dx e ( v ) = χ v f ( v ) for v ∈ Γ e ∈ E v j ) = e i α j f ( c + f ( c − j )    j ) = − e i α j f ′ ( c + f ′ ( c − j ) Now we consider λ n ( α ) as a function of α . Weyand Eigenvalues of Quantum Graphs

Φ ν ν Φ ν Φ ν Φ ν Φ ν Φ Nodal Surplus φ n = # of zeros of the n th eigenfunction ν n = # of subgraphs formed by removing the φ n zeros from Γ Nodal Surplus : φ n − ( n − 1) Nodal Deficiency : n − ν n Weyand Eigenvalues of Quantum Graphs

Φ ν Φ ν Φ ν Nodal Surplus φ n = # of zeros of the n th eigenfunction ν n = # of subgraphs formed by removing the φ n zeros from Γ Nodal Surplus : φ n − ( n − 1) Nodal Deficiency : n − ν n n = 3 , Φ 3 = 2, ν 3 = 3 n = 1, Φ 1 = 0, ν 1 =1 n = 2 , Φ 2 = 1, ν 2 = 2 Weyand Eigenvalues of Quantum Graphs

Nodal Surplus φ n = # of zeros of the n th eigenfunction ν n = # of subgraphs formed by removing the φ n zeros from Γ Nodal Surplus : φ n − ( n − 1) Nodal Deficiency : n − ν n n = 3 , Φ 3 = 2, ν 3 = 3 n = 1, Φ 1 = 0, ν 1 =1 n = 2 , Φ 2 = 1, ν 2 = 2 n = 1, Φ 1 = 0, ν 1 = 1 n = 2, Φ 2 = 1, ν 2 = 2 n = 3, Φ 3 = 2, ν 3 = 3 Weyand Eigenvalues of Quantum Graphs

Nodal Surplus φ n = # of zeros of the n th eigenfunction ν n = # of subgraphs formed by removing the φ n zeros from Γ Nodal Surplus : φ n − ( n − 1) Nodal Deficiency : n − ν n n = 3 , Φ 3 = 2, ν 3 = 3 n = 1, Φ 1 = 0, ν 1 =1 n = 2 , Φ 2 = 1, ν 2 = 2 n = 1, Φ 1 = 0, ν 1 = 1 n = 2, Φ 2 = 2, ν 2 = 2 n = 3, Φ 3 = 2, ν 3 = 3 Weyand Eigenvalues of Quantum Graphs

Morse Index Morse Index = # of negative eigenvalues of the Hessian matrix H i , j = d 2 λ n ( α ) d α i d α j 0 2 1 −0.2 1.8 0.8 0.6 −0.4 1.6 0.4 1.4 −0.6 0.2 1.2 −0.8 0 1 −1 −0.2 0.8 −1.2 −0.4 0.6 −1.4 −0.6 0.4 −0.8 −1.6 0.2 −1 1 −1.8 1 0 0 0 −1 1 −0.5 0.5 −2 0 0 0.5 1 −1 −1 −0.5 −1 −1 −0.5 0.5 1 0 0.5 −1 1 −0.5 0 Morse Index = 2 Morse Index = 0 Morse Index = 1 Weyand Eigenvalues of Quantum Graphs

Main Result Theorem (Berkolaiko & Weyand, 2013) Let λ n be a simple eigenvalue of H 0 whose eigenfunction has φ internal zeros. Consider the eigenvalues λ n ( α ) of H α as a function of α : α = (0 , 0 , . . . , 0) is a non-degenerate critical point of λ n ( α ) and the Morse index of this critical point is equal to φ − ( n − 1) Weyand Eigenvalues of Quantum Graphs

Partitions Proper m-Partition : Set of m points, none of which lie on vertices Partition Subgraphs : Subgraphs Γ j formed by applying Dirichlet conditions at the m-partition points Weyand Eigenvalues of Quantum Graphs

Corollary Λ( P ) := max j λ 1 (Γ j ) Equipartition : All partition subgraphs have the same first eigenvalue Weyand Eigenvalues of Quantum Graphs

Corollary Λ( P ) := max j λ 1 (Γ j ) Equipartition : All partition subgraphs have the same first eigenvalue Corollary (Berkolaiko & Weyand, 2013) Consider Λ on the set of equipartitions: the φ -equipartition formed from the zeros of the n th eigenfunction is a non-degenerate critical point of Λ and the Morse index of this critical point is equal to n − ν . Note : This strengthens the result of Band, Berkolaiko, Raz, and Smilansky (‘12) Weyand Eigenvalues of Quantum Graphs

Can one “hear” the shape of a graph? Given only eigenvalues, can one reconstruct the graph? Weyand Eigenvalues of Quantum Graphs

Can one “hear” the shape of a graph? Given only eigenvalues, can one reconstruct the graph? No, isospectral quantum graphs exist (Sunada, ’85). Cannot Determine: (Band and Parzanchevski, ’10) # of edges and vertices # of independent cycles ( β = | E | − | V | + 1) b 2a b a 2b a c c 2c Weyand Eigenvalues of Quantum Graphs

Only a Tree is a Tree On a tree, φ n = n − 1 ∀ n . Theorem (Band, 2013) If φ n = n − 1 ∀ n, then the graph is a tree. Weyand Eigenvalues of Quantum Graphs

References R. BAND, The nodal count { 0 , 1 , 2 , 3 , . . . } is a tree. preprint arXiv:1212.6710 [math-ph] , 2012. R. BAND, G. BERKOLAIKO, H. RAZ, AND U. SMILANSKY, The number of nodal domains on quantum graphs as a stability index of graph partitions , Comm. Math. Phys., 311 (2012), pp. 815-838. G. BERKOLAIKO AND T. WEYAND, Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions , Philosophical Transactions of the Royal Society A, accepted arXiv:1212.4475 [math-ph] , 2012. Contact Information Tracy Weyand www.math.tamu.edu/ ˜ tweyand tweyand@math.tamu.edu Weyand Eigenvalues of Quantum Graphs