swanson model: - beyond the pt-symmetry phase

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


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.



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



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)




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




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




H =P2

2m=

~ω4(A+A− − (A2

+ + A2−))

H =p2

2m− 1

2mω2Θ2

λ − iω

2[p,Θλ]


Marta

Marta

Marta




A+ = a†

A− ≈ a + λa†a +λ2

2a†2a


A+ ≈ a† +λ

2a†2 +

λ2

3a†3a

A− = a




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).


Marta



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).


Ψ(q) = eiα(q)φ(q), α(q) = i λ

12√

2q(6 + q2)

− ~2

2mϕ′′(q) +

~ω64λ2(2− q2)2ϕ(q) = Eϕ(q)


Marta



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.





A+ = a†

A− ≈ a + λa†a +λ2

2a†2a


A+ ≈ a† +λ

2a†2 +

λ2

3a†3a

A− = a




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

)]




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.


Marta



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)]

.



Nonstandard q-deformed harmonic oscillator.Ψ(q) = eiα(q)φ(q), α(q) = −i (η+6ζ)λ

12√

2(2ζ−η)q(3− q2)


− ~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


Marta


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 .




1 Nonstandard q-deformed oscillator⇒ non-hermitian hamitonian.2 Local gauge transformation⇒ V (q) =

∑cnqn



More recently...

Figura



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,



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).



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)

, .



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].




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

).




-1 0 1-2

-1

0

1

2

0

1003λ2 − 4r4 (1− λ4)

<ζ

η<

3λ2 + 4r4 (1− λ4)

r =

√1− 7λ4

16



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〉.




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〉.



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




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



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.




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)



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,




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



and now? ...

Figura



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



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).




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




Figura: Ω2 = ω2(1 − 4αβω2 ) Figura: m = ~

b20ω

1(1−α

ω− βω

)




-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




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|



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.



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.



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.



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〉.



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.




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),




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




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).



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

Marta

Marta

Marta

Marta


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).




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)



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 ′),



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.




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




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)




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) ,




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.



Open questions

1 Metric operators2 Exceptional Points?3 Time Evolution4 Work is in progress concerning Complex Scaling Method



Thanks!

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swanson model: - beyond the pt-symmetry phase

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