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XIX<vPHHQP<XX.UNLP-CONICET
Swanson Model:beyond the PT-symmetry phase.
Viviano Fernández, Romina Ramírez,Marta Reboiro.
Department of Physics-UNLP
Institute of Physics of La Plata-CONICET
July 1st 2021.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 1 / 61
XIX<vPHHQP<XX.UNLP-CONICET
1 A brief review of related previous works.2 no-standard hofp algebras and the Swanson Model.3 Swanson Model: regions in the parameter model-space.4 Swanson Model: spectrum and generalized eigenfunctions.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 2 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Some time ago...
PHYSICAL REVIEW C 72, 014305 (2005)
Nonstandard q-deformed realizations of the harmonic oscillator
A. Ballesteros,1 O. Civitarese,2 and M. Reboiro2,3
1Departamento de Fısica, Universidad de Burgos, Pza. Misael Banuelos, E-09001 Burgos, Spain2Department of Physics, University of La Plata, c.c. 67 1900, La Plata, Argentina
3Faculty of Engineering, University of Lomas de Zamora, C. C. Km 2, (1836) Lavallol, Argentina(Received 30 November 2004; published 14 July 2005)
The boson expansion method is applied to find the spectrum of a q-deformed harmonic oscillator. We use twodifferent boson expansions, each of them including a deformation parameter, defined in terms of exponential andlogarithmic functionals. The resulting Hamiltonians are bilinear forms of the transformed operators. Physicaleffects resulting from the deformation of the generators of the algebra are studied by comparing known finite-rangepotentials and the effective potentials obtained for each of the considered Hamiltonians.
DOI: 10.1103/PhysRevC.72.014305 PACS number(s): 21.60.Fw, 02.20.Uw
Figura
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 3 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Weyl Algebra
a†,a,N,1 :
[N,a] = −a[N,a†] = a†
[a,a†] = 1[1, ·] = 0
C = N − 12a†,a
Hopf Algebra: Unλ(h4)
A+,A−,N,M :
[N,A−] = −A−[N,A+] = −
(eλA+ − 1
)/λ
[A−,A+] = MeλA+
[M, ·] = 0
Cλ = NM − 12
eλA+−1
λ ,A−
Ballesteros A., Herranz F. J., Nieto L. M. and Negro J., J. Phys. A: Math. Gen. 33, 4859(2000)
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 4 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Exponential boson mapping
A+ = a†
A− = eλa†aN = eλA+−1
λ a
Logarithmic boson mapping
A+ = 1λ ln
(1
1−λa†
)a†
A− = aN = a†a
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 5 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
p = i
√m~ω
2(a† − a)
x =
√~
2mω(a† + a)
P = i√
m~ω2 (A+ − A−) ⇒ P = p − imωΘλ
X =√
~2mω (A+ + A−) ⇒ X = x + Θλ
Exponential Mapping Logarithmic Mapping
Θλ =√
~2mω
∑∞k=1
λk
k! a†λa Θλ = −√
~2mω
∑∞k=1
λk−1
k! a†λ
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 6 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
H =P2
2m=
~ω4(A+A− − (A2
+ + A2−))
H =p2
2m− 1
2mω2Θ2
λ − iω
2[p,Θλ]
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 7 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Exponential boson mapping
A+ = a†
A− ≈ a + λa†a +λ2
2a†2a
Logarithmic boson mapping
A+ ≈ a† +λ
2a†2 +
λ2
3a†3a
A− = a
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 8 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Hexp
(~ω/2)≈ p2 +
λ√2
(ip + q − ip(p2 + q2)
)+λ2
2
(p2 +
54
(p2 + q2)− p2 q2 − 34
p4 − 14
q4
−i12
p(p2 + q2 + 2)q),
Local gauge transformation:
Ψ(q) = eiα(q)φ(q), α(q) = −i λ
6√
2q(3− q2)
− ~2
2mϕ′′(q) +
~ω16λ2(1− q2)2ϕ(q) = Eϕ(q).
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 9 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Hlog
(~ω/2)≈ p2 +
λ√2
(ip − 1
2q − ip(ip − 1
2q)q − i
12
p3)
+λ2
8
(−11
4+
152
p2 − 72
q2 +112
p2q2 − 1912
p4 − 14
q4
+ip(−5p2 +73
q2 + 11)q).
Local gauge transformation:
Ψ(q) = eiα(q)φ(q), α(q) = i λ
12√
2q(6 + q2)
− ~2
2mϕ′′(q) +
~ω64λ2(2− q2)2ϕ(q) = Eϕ(q)
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 10 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
H =~ω4(ηA+A− + ζ(A2
+ + A2−))
H = g+(η, ζ)p2
2m+ ζ
12
mω2Θ2λ − i g+(η, ζ)
w2p,Θλ
+ g−(η, ζ)mω2
2(x2 + x ,Θλ)
− ωη
8(~−mω[x ,Θλ] + i [p,Θλ]) ,
g±(η, ζ) = (η ± 2ζ)/4.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 11 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Exponential boson mapping
A+ = a†
A− ≈ a + λa†a +λ2
2a†2a
Logarithmic boson mapping
A+ ≈ a† +λ
2a†2 +
λ2
3a†3a
A− = a
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 12 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Hexp
(~ω/4)≈ η
2(p2 + q2 − 1) + ζ(q2 − p2)
+λ√2
[η2
(3 + 3ip − q + p2q − ip(p2 + q2))
+ζ(3− 2q + p2q + ipq2 + ip3)]
+λ2
2
[η4
(−3 + 6p2 + 2ip(3− p2 − q2)q − p4 + q4)
−ζ(1 + 2(p2 + q2)− 2p2q2 − p4 − q4 − 4ipq
)]
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 13 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
Ψ(q) = eiα(q)φ(q), α(q) = −i λ
6√
2q(
3 (34ζ−η)(2ζ−η) − q2
)Local gauge transformation:
− ~2
8m(η − 2ζ)ϕ′′(q) + V (q)ϕ(q) = Eϕ(q).
V (q)
(~ω/4)= −η
2− (4ζ − η)2λ2
16(2ζ − η)− (ζ + η)√
2λq
+q2
8(4(η + ζ) + λ2(4ζ − η)) +
2ζ + η
2√
2λq3 − (2ζ − η)
16λ2q4.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 14 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
H ′log ≈ ~ω4
η2
(p2 + q2 − 1) + ζ(q2 − p2)
+λ
2√
2
[η2
(3 + 3ip − q + p2q − ip(p2 + q2))
+3ζ(1− ip + q − p2q − ipq2 + i13
p3)
]+λ2
12[η(−3 + 6p2 + 2ip(3− p2 − q2)q − p4 + q4)
−114ζ(−3 + 6(p2 − q2) + 6p2q2 − p4 − q4
+4ip(3− p2 + q2)q)]
.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 15 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.Ψ(q) = eiα(q)φ(q), α(q) = −i (η+6ζ)λ
12√
2(2ζ−η)q(3− q2)
Local gauge transformation:
− ~2
8m(η − 2ζ)ϕ′′(q) + V (q)ϕ(q) = Eϕ(q).
V (q)
(~ω/4)= −η
2− (6ζ + η)2
64(2ζ − η)λ2 − η
2√
2λq
+
(ζ +
η
2+
(6ζ + η)2
32(2ζ − η)λ2)
q2
+2ζ + η
4√
2λq3 − (6ζ + η)2
64(2ζ − η)λ2q4
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 16 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.Results
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4
-45
-40
-35
-30
-25
-20
-15
-10
VWS
VPT
V'log
Vlog
V'exp
Vexp
V(q
)
q
Woods-Saxon Potential:
V (r) =V0
1 + er−Ra0
,
Poeschl-Teller Potential:
VPT (r) =V0
(cosh ( rR ))
2 .
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 17 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Nonstandard q-deformed harmonic oscillator.
1 Nonstandard q-deformed oscillator⇒ non-hermitian hamitonian.2 Local gauge transformation⇒ V (q) =
∑cnqn
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 18 / 61
XIX<vPHHQP<XX.UNLP-CONICET
More recently...
Figura
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 19 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
Non-standard deformed Hamiltonian
Hλ = ηA+,A−+ ζ(A2+ + A2
−).
[N,A−] = −A−[N,A+] = −
(eλA+ − 1
)/λ
[A−,A+] = MeλA+
[M, ·] = 0
A+ = a†,
A− = δeλa†a + δβzeλa† ,
N =eλa† − 1
λa + β
eλa† + 12
M = δI,
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 20 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...Boson realization of Hλ
Hλ = (η − ζ)(p2 − i p,Θλ) + ζΘ2λ + (η + ζ)(x2 + x ,Θλ).
p =i√2
(a† − a),
x =1√2
(a† + a),
Θλ =1√2
∞∑k=1
λk
k !a†
ka.
O(λ3) : H ≈ H0 + Hres
H0 = −λ2ζ + 2(η + ζλ2)(a†a + 1/2) + λ(ηa† + ζa)
+ζa2 + (ζ + ηλ2/2)a†2,Hres = 2λ(ηa†2a + ζa†a2) + λ2(2ζa†2a2 + ηa†3a).
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 21 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...Quadratic Hamiltonian H0 (Swanson Model)
H0 = ω(a†a +12
) + αa2 + β(a†)2 + H00,
H0 = VHSW V−1, V = exp
(−a1 − a0
2x)
exp
(−i
a1 + a0
2p),
HSw = ω
(a†a +
12
)+ αa2 + βa†
2 − H00.
ω = 2η + 2ζλ2, α = ζ, β = λ2η2 + ζ, H00 = −5ζλ2/4,
a0 = − 2η2λ−2ζ2λ2(2η2−2ζ2+3ηζλ2)
, a1 = − 12(2η2−2ζ2+3ηζλ2)
, .
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 22 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model.- Z. Ahmed, Phys. Lett. A 294, (2002)287).- M. S. Swanson, J. Math. Phys. 45, (2004)585.- H F Jones, J. Phys. A: Math. Theor. 38, (2005)1741.- D.P. Musumbu, H.B. Geyer and W.D. Heiss, J. of Phys. A: Math.and Theor. 40, (2007)2.- A. Mostafazadeh, J. Phys. A Math Theor. 41, (2007)24.- P. Assis and A. Fring, J. Phys. A: Math. Theor. 41,(2008)244001.- C. Bender and H. Jones, J. Phys. A: Math. Theor. 41,(2008)244006.- Sinha and P. Roy, J. Phys. A: Math. Theor. 42, (2009)5.- M. Znojil, J. of Math. Phys. 50, (2009) 122105.- A. Mostafazadeh, Int.J.Geom.Meth.Mod.Phys.7, (2010)1191.- Ö. Yesiltas, J. Phys. A: Math. Theor. 44, (2011)30.- L. Inzunza and M.Plyushchay, arXiv:2104.08351v2[hep-th].
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 23 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
h = h†,h = WHSW W−1,W = exp
(−α−β4 p2
)⇒ h = ΥH0Υ−1,Υ = WV−1.
h = γa†,a+ %(a†2
+ a2)− H00,
γ =14
(ω + α + β +
√ω2 − 4αβω + α + β
),
% =14
(ω + α + β −
√ω2 − 4αβω + α + β
).
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 24 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...Eigenvalues and Eigenstates
H0 = H00 + Ω
(cd +
12
), Ω =
√D, D = ω2 − 4αβ.
d = g4 a− g2 a†,
c = −g3 a + g1 a†,[d , c
]= 1, c 6= d†.
H0|Φ〉 = En|Φ〉, En =
(n +
12
)Ω, |Φ〉 =
1√n!
cn|0d 〉.
|0d 〉 = Nd exp (τ a†2)|0〉, a|0〉 = 0, τ = (Ω− ω)/(4α).
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 25 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
-1 0 1-2
-1
0
1
2
0
1003λ2 − 4r4 (1− λ4)
<ζ
η<
3λ2 + 4r4 (1− λ4)
r =
√1− 7λ4
16
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 26 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
Pseudo-hermitian Hamiltonian.
h = ΥHΥ−1,h = h† ⇒ H†U = UH, U = Υ†Υ
h† = Υ−1†H†Υ†,
Bi-orthogonality.
H|ψ〉 = En|ψ〉, En ∈ RH†|ψm〉 = Em|ψm〉, |ψm〉 = U|ψm〉, 〈ψm|ψn〉 = δnm.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 27 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...Inner Product
〈 .|. 〉U : H×H → C, 〈ψ|φ〉U := 〈ψU|φ〉,
〈ψα|ψβ〉U = 〈ψα|ψβ〉 = δαβ , I =∑α
|ψα〉〈ψα| =∑α
|ψα〉〈ψα|
Mean Value Observables, o = o†.
O = Υ−1oΥ, o = o†.
〈ψ|O|φ〉U = 〈ψ|UO|φ〉 = 〈ψ|Υ†oΥ|φ〉.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 28 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
Time Evolution.
|I〉 =∑
k ck |Φk 〉 ⇒ |I(t)〉 = e−iHt |I〉 =∑
k
ck e−iEk t |Φk 〉,
|I〉 = U |I〉∑
k ck |Φk 〉 ⇒ |I(t)〉 = e−iH†t |I〉 =∑
k
ck e−iEk t |Φk 〉.
〈I(t)|O |I(t)〉 = 〈I(0)|UeiHtOe−iHt |I(0)〉 =
=∑n,m
cnc∗mei(Em−En)t〈Φm|Υ†oΥ|Φn〉.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 29 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...Uncertainty Relations and Squeezing.
∆2U(O) = 〈Φn|Υ† o2 Υ|Φn〉 − (〈Φn|Υ† o Υ|Φn〉)2.
-1 0 10.40.60.81.01.21.41.61.8
Q(x,p)Q(p,x)
Q(x,p)
Q(p,x)
X = Υ−1xΥ
P = Υ−1pΥ
Q(x ,p) = 2∆2UX,
Q(p, x) = 2∆2U P
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 30 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
Time Evolution: |I(0)〉 = |GZ 〉 = N∑ zn
γ√ρn
eiγEn |Φn〉.
-2 -1 0 1 2
-0.8
-0.4
0.0
0.4
0.8
<p>
<x> 0.0 0.5 1.0 1.5 2.00.6
0.8
1.0
1.2
1.4
1.6
1.8
2 p
2 x
t/T
4 x p
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 31 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...Wigner Function.
-2
0
2
(a)
00.050.100.150.200.250.300.35
-2
0
2 (b)
-2
0
2 (c)
-2
0
2 (d)
-2 0 2-2
0
2 (e)
x
p
W (x, p, t) =
1
2π
∫eipy 〈I(t)|Υ†|x −
y
2〉〈x +
y
2|Υ|I(t)〉dy,
∫W (x, p, t) dx dp = 1.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 32 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
General Hamiltonian.
H =ζλ2
2+ (η − ζλ2)(x2 + p2) + ζ(x2 − p2)
+ηλ2
4(x2 − p2)(x2 + p2)− i
ηλ2
4x ,p(x2 + p2)
+λη√
2(x − i p)(x2 + p2) +
λζ√2
(x2 + p2)(x + i p)
+ζλ2
2(x2 + p2)(x2 + p2)
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 33 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...Pseudo-hermitian hamiltonian.
h = ΥHΥ−1, Υ = e−F (p)e−G(x),
G(x) = g1(θ, λ)x2 + g2(θ, λ)x3 + g3(θ, λ)x4,
F (p) = f1(θ, λ)p2 − if2(θ, λ)p3 + f3(θ, λ)p4.
h = ΥHΥ−1:
h = h0(θ, λ) + h1(θ, λ)p2 + h2(θ, λ)x2 + h3(θ, λ)p4 +
h4(θ, λ)x4 + h5(θ, λ)x2,p2+ h6(θ, λ)x ,p2+
h7(θ, λ)x + h8(θ, λ)x3,
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 34 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Swanson model and deformed harmonic oscillator...
Time Evolution: |I(0)〉 = |GZ 〉 = N∑ zn
γ√ρn
eiγEn |Φn〉.
-2 -1 0 1 2
-0.8
-0.4
0.0
0.4
0.8
<p>
<x> 0.0 0.5 1.0 1.5
0
1
2
3
4
5
2 p
2 x
t/T x 103
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 35 / 61
XIX<vPHHQP<XX.UNLP-CONICET
and now? ...
Figura
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 36 / 61
XIX<vPHHQP<XX.UNLP-CONICET
PT-symmetry phase ...
Squeezing Hamiltonian
H = ~ω(
a†a +12
)+~α (a2 + a†
2),
Swanson Hamiltonian
H(ω, α, β) = ~ω(
a†a +12
)+~α a2 + ~β a†
2.
ω²-4 α β>0
α= β
-2 -1 0 1 2
-2
-1
0
1
2
α
ω
β
ω
Figura: PT-symmetry phase
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 37 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
x = b0√2
(a† + a
)p = i~√
2b0
(a† − a
)H(ω, α, β) & Hc(ω, α, β):
H(ω, α, β) =12~(ω + α + β)
(xb0
)2
+ ~(α− β)
2
(2 x
i~
p + 1)
+12~(ω − α− β)
(b0 p~
)2
Hc(ω, α, β) = H(ω, β, α).
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 38 / 61
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Beyond PT-symmetry phase...
H = 12m P2φ(X ) + k
2 X 2φ(X )
P =
(p + i~
α− β(ω − α− β)b2
0x),
X = x ,
m = m(ω, α, β,b0) =~
(ω − α− β)b20
Ω = Ω(ω, α, β) =√ω2 − 4αβ = |Ω|eiφ,
k = m Ω2
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 39 / 61
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Beyond PT-symmetry phase...
Figura: Ω2 = ω2(1 − 4αβω2 ) Figura: m = ~
b20ω
1(1−α
ω− βω
)
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 40 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
-Region I: m > 0 and Ω2 > 0-Region III: m < 0 and Ω2 > 0-Region II: m > 0 and Ω2 < 0-Region IV: m < 0 and Ω2 < 0
(I) m>0 & Ω²>0
m>0 & Ω²<0
(II)
(III)
m<0 & Ω²>0
m<0 & Ω²<0
(IV)
-2 -1 0 1 2
-2
-1
0
1
2
α
ω
β
ω
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 41 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
Defining...
σ =
(mΩ
~
)1/2
b0 = eiγ± |σ|.
Ω = eiφ|Ω|.
sg(m) sg(Ω2) γ φI + + 0 0
π/2 πIII - + π/2 0
0 πII + - π/4 π/2
−π/4 −π/2IV - - −π/4 π/2
π/4 −π/2Ω = eiφ|Ω|
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 42 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase.
Region II: Parabolic Barrier
D. Chruscinski, J. Math. Phys. 44, (2003), 3718.
D. Chruscinski, J. Math. Phys. 45, (2004), 841.
G. Marucci and C. Conti, Phys. Rev. A 94, (2016), 052136.
Region III: Harmonic Oscillator with effective negative mass
M. Znojil, P. Siegl, G. Lévai, Phys. Lett. A 373 (2009) 1921.
E. S. Polzik and K. Hammerer Ann. Phys. (Berlin) 527, (2015) A15.
F. Di Mei et al., Phys. Lett. 116, (2016)153902.
M. A. Khamehchi et al., Phys. Lett. 118, (2017)155301.
J. Kohler et al.,Phys. Rev. Lett. 120, (2018)013601.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 43 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Rigged Hilbert Space...
Gelfand Triplet...
Φ ⊂ H ⊂ Φ×
Φ dense in HH: Hilbert SapceΦ× = F | F : Φ→ C, F (φ) = 〈φ|F 〉
Φ depends on the problem, i.e. D or S.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 44 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Rigged Hilbert Space.
Example: f±n generalized functions.D. Chruciski, J. Math. Phys. 44 (2003)3718.
H× = − ζ2
(uv + v u) = iζ(
uddu
+12
), [u, v ] = i
uf−0 = 0 v f +0 = 0, H×f±0 = ±i ζ2 f±0
f−0 (u) = δ(u) f +0 (u) = 1.
f−n = (−i)n√
n!vnf−0 f +
n = 1√n!
unf +0 , H×f±n = ±iζ(n + 1
2 )f±nf−n (u) = (−i)n
√n!δ(n)(u) f +
n (u) = (1)n√
n!un.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 45 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Rigged Hilbert Space.
Properties of f±n ∫ ∞−∞
f +n (u)fm(u)du = δnm
∞∑n=0
f +n (u)fn(u′) = δ(u − u′).
For φ ∈ D:
|φ〉 =∞∑
n=0
|f +n 〉〈f−n |φ〉.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 46 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Rigged Hilbert Space.Continuous Spectrum:
uddu
ΨE± = −
(iEζ
+12
)ΨE±,
ΨE±(u) =
1√2πζ
u−(i E
ζ+ 12 )
±
with the distribution:
sλ+ =
sλ s ≥ 00 s < 0
sλ− =
0 s ≥ 0|s|λ s < 0
∫ ∞−∞
ΨE1± (u)
∗ΨE2± (u)du = δ(E1 − E2)∫ ∞
−∞ΨE±(u)
∗ΨE±(u′)dE = δ(u − u′).
φ(u) =∑±
∫ΨE±(u)〈ΨE
±(u)|φ〉dE.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 47 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
Gauge Transformation.
H× φ(x) = E φ(x).
φ(x) = eα−β
ω−α−βx2
2b20 f (x),
y = y(x , γ) = |σ|eiγ xb0,
~Ω12(p2
y + y2) f (y) = Ef (y),
H×c ψ(x) = Ec ψ(x).
ψ(x) = eβ−α
ω−α−βx2
2b20 g(x),
py = iddy.
~Ω12(p2
y + y2)g(y) = Ecg(y),
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 48 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
Region II.
Ω = ±i|Ω|
σ = b0
√mΩ
~= e±iπ/4|σ| ⇒ γ = ±π/4.
I
II
III
IV
-2 -1 0 1 2
-2
-1
0
1
2
α
ω
β
ω
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 49 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
Swanson model
12
(p2
y + y2)
f (y) = E f (y), 12
(p2
y + y2)
g(y) = Ec g(y),
u = u(γ) =y−ipy√
2, v = v(γ) =
y+ipy√2
u = u(−γ) =y−ipy√
2, v = v(−γ) =
y+ipy√2
u∗ = v , v∗ = u.
~Ω2 (uv + vu) f (u) = E f (u), ~Ω
2 (uv + vu) g(u) = Ec g(u).
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 50 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...~Ω2 (uv + vu) f = E fγ = π/4 :
f +n (u) = 1√
n!un En = iE0[n]
f−n (u) = (−1)n√
n!δ(n)(u) En = −iE0[n]
γ = −π/4 :
f +n (u) = 1√
n!un En = −iE0[n]
f−n (u) = (−1)n√
n!δ(n)(u) En = iE0[n]
〈g±m (u)|f±n (u)〉 = δnm
E0 = ~|Ω|,
~Ω2 (uv + vu) g = Ec g.γ = π/4 :
g+n (v) = (−1)n
√n!δ(n)(v) Ec
n = iE0[n]
g−n (v) = 1√n!
vn Ecn = − iE0[n]
γ = −π/4 :
g+n (v) = (−1)n
√n!δ(n)(vn) En = −iE0[n]
g−n (v) = 1√n!
vn En = iE0[n]
〈g±m (v)|f±n (v)〉 = δnm
[n] = n + 1/2.Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 51 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase.
S(y ,u) = 12 y2 −
√2yu + 1
2 u2, Sc(yc , vc) = 12 y2
c −√
2ycvc + 12 v2
c .
∂S∂y = py ,
∂S∂u = −v , ∂Sc
∂yc= pyc ,
∂Sc∂vc
= −uc .
f +n (y) = C
∫Γ
f +n (u)eS(y,u)du, g−n (yc) = C
∫Γ
g−n (vc)e−Sc(yc ,vc)dvc ,
g+n (y) = f +
n (y)∗, f−n (y) = g−n (y)
∗.
C∫
Γ
eS(y,u)−Sc(y ′c ,vc)∗du = C δ(y − y ′), vc = vc(−γ).
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 52 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
E±n = ±i~ |Ω|(
n +12
),
y(x , γ) = eiγ± xb0|σ|,
f±n (y) ∝ 2−n2 e−
y2
2 |σ|2Hn (y) ,
Ec±n = ∓i~ |Ω|
(n +
12
),
yc(x , γ) = e−iγ± xb0|σ|,
g±n (yc) ∝ 2−n2 e−
y2c2 Hn (yc) ,
〈Ψ±n |φ±m〉 = δmn,
∞∑n = 0σ = ±
ψσn (x)∗φσm(x ′) = δ(x − x ′).
φ±m (x) = e
α−βω−α−β
x22b2
0 f±m (x), ψ±n (x) = e
β−αω−α−β
x22b2
0 g±n (x)
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 53 / 61
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Beyond PT-symmetry phase...Region II: Continuous Spectrum.
f ν±(u) = uν±, f ν±(u)∗ = vνc±,
〈gν′± (vc)|f ν±(u)〉 = δνν′
f E+ (y) = C
∫Γ
uνeS(y,u)du
= CΓ(ν + 1) D−ν−1(−i√
2y)
f E−(y) = CΓ(ν + 1) D−ν−1(i
√2y)
gν±(vc)∗ = u−(ν+1)± , gν±(vc) = v−(ν+1)
c± .
ν = −i E~|Ω| , ν∗ = −(ν + 1).
gE+(yc) = C
∫Γ
vνc e−S(yc ,vc)dvc
= C Γ(ν + 1) D−ν−1(−√
2yc)
gE−(yc) = C Γ(ν + 1) D−ν−1(
√2yc),∫ ∞
−∞gE±(y∗(x))
∗f E ′± (y(x))dx = δ(E − E ′),∫ ∞
−∞gE±(y∗(x))
∗f E±(y(x))dE = δ(x − x ′),
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 54 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...Region II: Continuous Spectrum.
f E±(y) ∝ Γ(ν + 1) D−ν−1(∓i
√2y), gE
±(y∗)∗ ∝ Γ(−ν) Dν(∓
√2y)
Poles of f E± → En = −i~Ω(n + 1/2), f +
n → e−iEnt = e−~Ω(n+1/2)t
Poles of gE±∗ → En = +i~Ω(n + 1/2), f−n → e−iEnt = e~Ω(n+1/2)t
(Poles Γ(λ), λ = −n.)
Φ− = φ−|φ− = 〈φ|gE±〉 ∈ H−, Φ+ = φ+|φ+ = 〈φ|f E
±〉 ∈ H+
φ−(x , t) =∑
n
e+~Ω(n+1/2)t〈φ−|f 〉∗f +
n (x)n
φ+(x , t) =∑
n
e−~Ω(n+1/2)t〈φ+|f +n 〉∗f−n (x)
H = L2, Φ = S.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 55 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
Region III:
Ω = ±|Ω|, m = −|m|
σ = b0
√mΩ
~= eiγ |σ| ⇒ γ = 0, π/2.
I
II
III
IV
-2 -1 0 1 2
-2
-1
0
1
2
α
ω
β
ω
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 56 / 61
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Beyond PT-symmetry phase...
γ = 0 :
En = −~|Ω|(
n +12
)f +n (y) ∝ e−
y2
2 Hn(y)
g+n (y) = f +
n (y)∗.
γ = π/2 :
En = ~|Ω|(
n +12
)f−n (y) ∝ e−
y2
2 Hn(y)
g−n (y) = f−n (y)∗.
φ±m (x) = e
α−βω−α−β
x22b2
0 f±m (x), ψ±n (x) = e
β−αω−α−β
x22b2
0 g±n (x)
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 57 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
ω = α + β, α. 6= β.
H×(θ) = ~(α + β)
(xb0
)2
+ ~(α− β)
2
(2 x
i~
p + 1),
H×c (θ) = ~(α + β)
(xb0
)2
+ ~(β − α)
2
(2 x
i~
p + 1).
φ(x) = e− x2
4b20
α+βα−β x−
12 + E
~(α−β) ψ(x) = ex2
4b20
α+βα−β x−
12−
Ec~(α−β) ,
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 58 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Beyond PT-symmetry phase...
Free particle: k = 0 (Ω2 = ω2 − 4αβ = 0).
− ~2
2md2φ(x)
dx2 = E φ(x),
the wave function can be written as φ(x) = Aeikx + Ae−ikx , withk =
√2ε
~(ω−α−β)b20.
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 59 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Open questions
1 Metric operators2 Exceptional Points?3 Time Evolution4 Work is in progress concerning Complex Scaling Method
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 60 / 61
XIX<vPHHQP<XX.UNLP-CONICET
Thanks!
email:[email protected]
Viviano Fernández, Romina Ramírez,Marta Reboiro. UNLP-CONICET Swanson Model July 1st 2021. 61 / 61