Zeta Functions of the Dirac Operator on Quantum Graphs Tracy Weyand - PowerPoint PPT Presentation

Zeta Functions of the Dirac Operator on Quantum Graphs Tracy Weyand Baylor University Waco, TX 76798-7328 tracy weyand@baylor.edu Joint work with Jon Harrison and Klaus Kirsten QMath13: Mathematical Results in Quantum Physics October 10, 2016

Metric Graphs Γ = { V , B , L } Quantum Graph: Metric Graph + Differential Operator T. Weyand Zeta Functions

Dirac Operator D = − i α d + m β d x b α and β are 4x4 matrices that satisfy α 2 = β 2 = I and αβ + βα = 0 Vertex Conditions: A ψ + + B ψ − = 0 ψ + = ( ψ 1 1 (0) , ψ 1 2 (0) , . . . , ψ B 2 (0) , ψ 1 1 ( L 1 ) , ψ 1 2 ( L 1 ) , . . . , ψ B 1 ( L B ) , ψ B 2 ( L B )) T ψ − = ( − ψ 1 4 (0) , ψ 1 3 (0) , ψ 1 4 ( L 1 ) , − ψ 1 3 (0) , . . . , ψ B 3 ( L 1 ) , . . . , ψ B 4 ( L B ) , − ψ B 3 ( L B )) T The operator is self-adjoint if and only if A and B are 4 B x 4 B matrices that satisfy rank( A , B ) = 4 B and AB † = BA † . [1] J. Bolte and J. M. Harrison, Spectral statistics for the Dirac operator on graphs, J. Phys. A: Math. Gen. 36:2747 (2003). T. Weyand Zeta Functions

Solutions D = − i α d + m β d x b � 0 � � I � σ 2 0 α = β = σ 2 0 0 − I Solutions to D ψ k = E ( k ) ψ k are of the form         1 0 1 0 0 1 0 1 ψ b ( x b ) = µ b    e i kx b + µ b    e i kx b + ˆ µ b    e − i kx b + ˆ µ b    e − i kx b α     α     0 β − i γ ( k ) 0 β i γ ( k )     i γ ( k ) 0 − i γ ( k ) 0 where γ ( k ) := E ( k ) − m � k 2 + m 2 E ( k ) := k T. Weyand Zeta Functions

Solutions Solutions to D ψ k = E ( k ) ψ k are of the form         1 0 1 0 0 1 0 1 ψ b ( x b ) = µ b  e i kx b + µ b  e i kx b + ˆ µ b  e − i kx b + ˆ µ b  e − i kx b         α   β   α   β   0 − i γ ( k ) 0 i γ ( k )     i γ ( k ) 0 − i γ ( k ) 0 where γ ( k ) := E ( k ) − m � k 2 + m 2 E ( k ) := k Solutions to D ψ k = − E ( k ) ψ k are of the form         i γ ( k ) 0 − i γ ( k ) 0 0 − i γ ( k ) 0 i γ ( k )  e i kx b + µ b  e i kx b + ˆ  e − i kx b + ˆ ψ b ( x b ) = µ b     µ b   µ b   e − i kx b  α   β   α   β   0 1 0 1     1 0 1 0 T. Weyand Zeta Functions

Secular Equation For positive solutions: � cot kL � − csc kL �� det A + γ ( k ) B = 0 − csc kL cot kL For negative solutions: � cot kL � �� − csc kL det γ ( k ) A − B = 0 − csc kL cot kL T. Weyand Zeta Functions

Spectral Zeta Function Given the set of roots { . . . < k − 2 < k − 1 < k 1 < k 2 < . . . } of the secular equation, the spectral zeta function is defined as ∞ ′ E ( k j ) − s � ζ ( s ) = 2 j = −∞ ∞ ′ k − s � = 2 j j = −∞ in the massless case. T. Weyand Zeta Functions

Rose Graph T. Weyand Zeta Functions

Vertex Conditions u b o ✈ b (0) = u b t ✈ b ( L b ) = η for all bonds b B B � � u b o ✇ b (0) = u b t ✇ b ( L b ) b =1 b =1 where � ψ b � − ψ b � � 1 ( x b ) 4 ( x b ) ✈ b ( x b ) = ✇ b ( x b ) = and ψ b ψ b 2 ( x b ) 3 ( x b ) Secular Equation: B cos θ b − cos kL b where cos θ b = 1 � t ) − 1 ) 2 tr ( u b o ( u b = 0 sin kL b b =1 T. Weyand Zeta Functions

Spectral Zeta Function B cos θ b − cos zL b � f ( z ) = z sin zL b b =1 ∞ ′ k − s � ζ ( s ) = 2 j j = −∞ z − s f ′ ( z ) = 1 � f ( z ) d z i π C = 1 � z − s d d z log f ( z ) d z i π C where C is any contour that encloses the zeros of f (while avoiding its poles). [2] J. Harrison and K. Kirsten, Zeta functions of quantum graphs, J. Phys. A: Math. Theor. 44 (2011). T. Weyand Zeta Functions

Contours i) ii) α α Contour C Contour C ′ The shaded circles are the zeros of f and the empty circles are the poles. T. Weyand Zeta Functions

Spectral Zeta Function – Rose Graph ζ ( s ) = ζ p ( s ) + ζ l ( s ) + ζ b ( s ) B � − 1 ∞ � − s � � − s � n π � n π � � � ζ p ( s ) = 2 + L b L b n = −∞ n =1 b =1 � π B � − s = 2( e − i π s + 1) ζ R ( s ) � L b b =1 ζ l ( s ) = 0 if Re( s ) > 0 T. Weyand Zeta Functions

Spectral Zeta Function – Rose Graph � ∞ ζ b ( s ) = e i ( π − α ) s 2 sin π s u − s d d u log f ( ue i α ) du π 0 This converges for 0 < Re( s ) < 2. � ∞ ζ b ( s ) = e i ( π − α ) s 2 sin π s u − s d � ue i α ˆ � d u log f ( u ) du π 0 B cos θ b − cos L b e i α u ˆ � f ( u ) = sin L b e i α u b =1 T. Weyand Zeta Functions

Spectral Zeta Function – Rose Graph Theorem �� 1 ζ ( s ) = e i ( π − α ) s 2 sin s π d u + 1 u − s d � ue i α ˆ � d u log f ( u ) π s 0 � π � ∞ B � − s u − s d � + 2( e − i π s + 1) ζ R ( s ) d u log ˆ � + f ( u ) d u L b 1 b =1 where Re( s ) < 2 and B cos θ b − cos L b e i α u ˆ � f ( u ) = . sin L b e i α u b =1 T. Weyand Zeta Functions

Spectral Determinant – Rose Graph ∞ ′ k 2 det ′ ( D ) = � j j = −∞ = exp( − ζ ′ (0)) � B � 2 B � 2 = (2 π ) 2 B ( − 1) B +1 � L b cos θ b − 1 � � B 2 L b π b =1 b =1 T. Weyand Zeta Functions

Spectral Zeta Function – General Graph Without Mass Theorem �� 1 ζ ( s ) = e i ( π − α ) s 2 sin s π d u + 4 B − 1 u − s d � ( ue i α ) 4 B − 1 ˆ � d u log f ( u ) π s 0 � π � ∞ B � − s u − s d � + 2( e − i π s + 1) ζ R ( s ) d u log ˆ � + f ( u ) d u L b 1 b =1 where Re( s ) < M and � cot ue i α L − csc ue i α L � �� ˆ f ( u ) = det A + B . − csc ue i α L cot ue i α L T. Weyand Zeta Functions

Spectral Determinant – General Graph Without Mass B c 02 ( − 1) B � det ′ ( D ) = (2 L b ) 2 det( A − i B ) 2 b =1 c 0 = f (0) � = 0 T. Weyand Zeta Functions

Spectral Zeta Function – General Graph With Mass Given the set of roots { k 1 , k 2 , . . . } of the positive energy secular equation and the set of roots { ˜ k 1 , ˜ k 2 , . . . } to the negative energy secular equation, the spectral zeta function is defined as ∞ ∞ � − s ′ E ( k j ) − s + 2 ′ � � � − E ( ˜ ζ ( s ) = 2 k j ) j =1 j =1 ∞ ∞ � − s � − s � � 2 + m 2 �� � � ˜ k 2 j + m 2 = 2 + 2 − k j j =1 j =1 = ζ + ( s ) + ζ − ( s ) . T. Weyand Zeta Functions

General Graph With Mass Positive Eigenvalues: � cot zL � − csc zL �� f ( z ) = det A + γ ( z ) B − csc zL cot zL ζ + ( s ) = 1 � ( z 2 + m 2 ) − s / 2 d d z log f ( z ) d z i π C Negative Eigenvalues: � cot zL � �� − csc zL g ( z ) = det γ ( z ) A − B − csc zL cot zL ζ − ( s ) = ( − 1) − s � ( z 2 + m 2 ) − s / 2 d d z log g ( z ) d z i π C T. Weyand Zeta Functions

Contour i) ii) i m i m α α − i m − i m Contour C Contour C ′ The shaded circles are the zeros of f / g and the empty circles are the poles. T. Weyand Zeta Functions

Spectral Zeta Function – General Graph With Mass � − s / 2 B ∞ �� n π � 2 � � ζ + + m 2 p ( s ) = 2 L b n =1 b =1 � π B � � 2 � � − s � mL b s � = 2 E 2 , π L b b =1 � � ∞ b ( s ) = 2 � π s ( t 2 − m 2 ) − s / 2 d ζ + π sin d t f ( i t ) d t 2 m which converges for − 1 < Re( s ) < 1. T. Weyand Zeta Functions

Spectral Zeta Function – General Graph With Mass Theorem � π B � − s � � 2 � s � mL b � ζ ( s ) = 2(1 + ( − 1) − s ) 2 , E π L b b =1 � �� ∞ + 2 � π s ( t 2 − m 2 ) − s / 2 d d t log ˆ π sin f ( t ) d t 2 m � ∞ � ( t 2 − m 2 ) − s / 2 d +( − 1) − s d t log ˆ g ( t ) d t m where − 1 < Re( s ) < 1 and � coth tL � �� − csch tL ˆ f ( t ) = det A + ˆ γ ( t ) B − csch tL coth tL � � coth tL − csch tL �� g ( t ) = det ˆ γ ( t ) A − B ˆ , and − csch tL coth tL √ t 2 − m 2 + i m ˆ γ ( t ) = . t T. Weyand Zeta Functions

Summary We found a formulation of the spectral zeta function of the Dirac operator using a contour integral technique. In the case of zero mass, we analytically continued our expression to a domain including s = 0 and calculated the zeta-regularized spectral determinant. We did this first for a rose graph without mass, and then for a general graph with and without mass. T. Weyand Zeta Functions

References [1] J. Bolte and J. M. Harrison, Spectral statistics for the Dirac operator on graphs, J. Phys. A: Math. Gen. 36(11):2747-2769 (2003). [2] J. Harrison and K. Kirsten, Zeta functions of quantum graphs, J. Phys. A: Math. Theor. 44(33):235301, 29 (2011). [3] J. Harrison, T. Weyand, and K. Kirsten, Zeta functions of the Dirac Operator on quantum graphs, J. Math. Phys. 57(10):102301, 17 (2016). T. Weyand Zeta Functions

T. Weyand Zeta Functions