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Asymptotic and Numerical Study of Resonant Tunneling in Quantum Waveguides with Several Equal Resonators Oleg Sarafanov Differential Operators on Graphs and Waveguides Graz, Austria, February 25 – March 1, 2019 Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

Statement of the problem Figure: Waveguide G ♣ ε q The wave function u of a free electron of energy k 2 satisfies the boundary value problem ✁ ✁ ∆ ✁ k 2 ✠ u ♣ x , y ; ε q ✏ 0 , in G ♣ ε q ; u ♣ x , y ; ε q ✏ 0 , on ❇ G ♣ ε q . Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

Radiation conditions Auxiliary boundary value problem in the cross-section of the strip ✡ 2 ✁ ❇ 2 ✂ π q ✁ ❇ y 2 ✁ λ 2 ✠ λ 2 Ψ ♣ y q ✏ 0 , q ✏ , ñ L Ψ ♣✁ L ④ 2 q ✏ Ψ ♣ L ④ 2 q ✏ 0 . q ✏ 1 , 2 , . . . . We assume that 1 ➔ k 2 ➔ λ 2 λ 2 2 , and set the radiation conditions ★ e i ν x Ψ 1 ♣ y q � S 11 ♣ k ; ε q e ✁ i ν x Ψ 1 ♣ y q � O ♣ e δ x q , x Ñ ✁✽ ; u ♣ x , y ; ε q ✏ S 12 ♣ k ; ε q e i ν x Ψ 1 ♣ y q � O ♣ e ✁ δ x q , x Ñ �✽ ; ❝ 1 L ν cos π y ❜ k 2 ✁ λ 2 δ → 0 , ν ✏ 1 , Ψ 1 ♣ y q ✏ L . Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

Resonant tunneling Reflection coefficient and transition coefficient R ♣ k ; ε q ✏ | S 11 ♣ k ; ε q | 2 , T ♣ k ; ε q ✏ | S 12 ♣ k ; ε q | 2 . We have R ♣ k ; ε q � T ♣ k ; ε q ✏ 1 . We are interested in 1 The asymptotics of resonant energies k 2 r ♣ ε q as ε Ñ 0; 2 The heights T ♣ k r , ε q of resonant peaks; 3 The width Υ ♣ ε q of resonant peaks at half-height. Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

Single resonator — reference Figure: Waveguide G ♣ ε q with a single resonator L. Baskin, P. Neittaanm¨ aki, B. Plamenevskii, and O. Sarafanov. Resonant Tunneling: Quantum Waveguides of Variable Cross-Section, Asymptotics, Numerics, and Applications // Lecture Notes on Numerical Methods in Engineering and Sciences, Springer, 2015, 275 pp. Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

Single resonator — asymptotics The results obtained: e ✁ Q ε 2 π ④ ω � O ♣ ε 2 π ④ ω � 2 ✁ δ q , k 2 r ♣ ε q ✏ k 2 1 ✁ ✠ 1 � O ♣ ε 2 ✁ δ q T ♣ k ; ε q ✏ , ☛ 2 k 2 ✁ k 2 ✄ r 1 � P 2 ε 4 π ④ ω Υ ♣ ε q ✏ 2 P ε 4 π ④ ω . Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

n equal resonators — resonant energies Figure: Waveguide G ♣ ε q with n equal resonators The resonant energies e ✁ Q j ε 2 π ④ ω � O ♣ ε 4 π ④ ω q , k 2 r , j ♣ ε q ✏ k 2 j ✏ 1 , . . . , n ; π j Q j ✏ A � 2 B cos n � 1 . Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

n equal resonators — peaks The transition coefficient and the widths of peaks 1 ✁ ✠ 1 � O ♣ ε 4 π ④ 3 ω q T ♣ k ; ε q ✏ ; ☛ 2 j ✏ 1 ♣ k 2 ✁ k 2 ✄➧ n r , j q 1 � P 2 ♣ n q ε 2 ♣ n � 1 q π ④ ω ✂ π j 4 ✡ ♣ n � 1 q P ♣ n q B n ✁ 1 sin 2 ε 4 π ④ ω . Υ j ♣ ε q ✏ n � 1 Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

Numerics for one resonator 31 30 29 res k 2 28 27 26 num asym 25 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ǫ Figure: Resonant energy 3 2.5 2 Υ 1.5 1 0.5 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ǫ Figure: Width of a resonant peak Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

Numerics for 10 resonators 1 0.8 0.6 T 1 0.4 0.2 0 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 k 2 -k 2 res,middle Figure: Resonant peaks for ε ✏ 0 . 1 1 0.8 0.6 T 1 0.4 0.2 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 k 2 -k 2 res,middle Figure: Resonant peaks for ε ✏ 0 . 3 Asymptotic and Numerical Study of Resonant Tunneling in Quantum Oleg Sarafanov

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