Nuclear Physics Institute ASCR Reˇz, Czech...
Transcript of Nuclear Physics Institute ASCR Reˇz, Czech...
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Non-Hermitian operatorsin quantum physics
David KREJCIRIK
http://gemma.ujf.cas.cz/ david/
Nuclear Physics Institute ASCR
Rez, Czech Republic
First, there’s the room you can see through the glass – that’s just the same as our
drawing-room, only the things go the other way.
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Hors de ligne
(Outline)
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Hors de ligne
(Outline)
1. QM with non-Hermitian operators
(just some conceptual remarks)
2. PT-symmetry
(what is known and my point of view)
3. physical PT-symmetric models in QM
(non-self-adjoint Robin boundary conditions)
4. imaginary cubic oscillator
(about the non-existence of the metric operator)
5. Conclusions
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¿ QM with non-Hermitian operators ?
R I
H∗ = H HPT = H
C
Imaginary Numbers by Yves Tanguy, 1954
(Museo Thyssen-Bornemisza, Madrid)
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Insignificant non-Hermiticity
Example 1. evolution operator U(t) = exp(−itH) :
iU(t) = H U(t)
U(0) = I
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Insignificant non-Hermiticity
Example 1. evolution operator U(t) = exp(−itH) :
iU(t) = H U(t)
U(0) = I
Example 2. resolvent operator R(z) = (H − z)−1, z ∈ C
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Insignificant non-Hermiticity
Example 1. evolution operator U(t) = exp(−itH) :
iU(t) = H U(t)
U(0) = I
Example 2. resolvent operator R(z) = (H − z)−1, z ∈ C
Theorem (spectral theorem).
Let H = H∗. Thenf(H) =
∫
σ(H)
f(λ) dEH(λ)
for any complex-valued continuous function f .
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Technical non-Hermiticity
X X XX XX
X X XX XX
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Technical non-Hermiticity
Example 1. complex scaling Hθ := Sθ(−∆+ V )S−1θ , (Sθψ)(x) := eθ/2 ψ(eθx)
θ = 0 ℑθ > 0
X X XX XX
X X XX XX
[Aguilar/Balslev, Combes 1971], [Simon 1972], [Van Winter 1974], . . .
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Technical non-Hermiticity
Example 1. complex scaling Hθ := Sθ(−∆+ V )S−1θ , (Sθψ)(x) := eθ/2 ψ(eθx)
θ = 0 ℑθ > 0
X X XX XX
X X XX XX
[Aguilar/Balslev, Combes 1971], [Simon 1972], [Van Winter 1974], . . .
Example 2. adiabatic transition probability for H(t) := ~γ(t/τ) · ~σ, τ →∞
[Berry 1990], [Joye, Kunz, Pfister 1991], [Jaksic, Segert 1993], . . .
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Technical non-Hermiticity
Example 1. complex scaling Hθ := Sθ(−∆+ V )S−1θ , (Sθψ)(x) := eθ/2 ψ(eθx)
θ = 0 ℑθ > 0
X X XX XX
X X XX XX
[Aguilar/Balslev, Combes 1971], [Simon 1972], [Van Winter 1974], . . .
Example 2. adiabatic transition probability for H(t) := ~γ(t/τ) · ~σ, τ →∞
[Berry 1990], [Joye, Kunz, Pfister 1991], [Jaksic, Segert 1993], . . .
Example 3. Regge theory Hl := −d2
dr2+l(l + 1)
r2+ V (r), l ∈ C
[Regge 1957], [Connor 1990], [Sokolovski 2011], . . .
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Approximate non-Hermiticityopen systems
Example 1. radioactive decay
Example 2. dissipative Schrodinger operators in semiconductor physics
Baro, Behrndt, Kaiser, Neidhardt, Rehberg, . . .
Example 3. repeated interaction quantum systems
Bruneau, Joye, Merkli, Pillet, . . .
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¿ Fundamental non-Hermiticity ?
i.e. non-Hermitian observables,
without violating physical axioms of QM
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¿ Fundamental non-Hermiticity ?
i.e. non-Hermitian observables,
without violating physical axioms of QM
¡ no !Theorem (Stone’s theorem).
Unitary groups on a Hilbert space are generated by self-adjoint operators.
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¿ Fundamental non-Hermiticity ?
i.e. non-Hermitian observables,
without violating physical axioms of QM
¡ no !Theorem (Stone’s theorem).
Unitary groups on a Hilbert space are generated by self-adjoint operators.
¿ yes ?
by changing the Hilbert space,
preserving a similarity to self-adjoint operators
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Non-Hermitian Hamiltonians with real spectra
−∆+ V in L2(R)
V (x) = x2 + ix3 [Caliceti, Graffi, Maioli 1980]
−1 0 1 2 3ε
1
3
5
7
9
11
13
15
17
19E
ne
rgy
V (x) = x2 (ix)ε
[Bessis, Zinn-Justin][Bender, Boettcher 1998]
[Dorey, Dunning, Tateo 2001][Shin 2002]
[Azizov, Kuzhel, Gunther, Trunk 2010]
V (x) =
i sgn(x) if x ∈ (−L,L)
∞ elsewhere[Znojil 2001]
¿ What is behind the reality of the spectrum ?
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PT - symmetry[H,PT ] = 0
(Pψ)(x) := ψ(−x)
(Tψ)(x) := ψ(x)
We have in mind H = −∆+ V on L2(Rd) with V (−x) = V (x).
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PT - symmetry[H,PT ] = 0
(Pψ)(x) := ψ(−x)
(Tψ)(x) := ψ(x)
We have in mind H = −∆+ V on L2(Rd) with V (−x) = V (x).
PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H)
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PT - symmetry[H,PT ] = 0
(Pψ)(x) := ψ(−x)
(Tψ)(x) := ψ(x)
We have in mind H = −∆+ V on L2(Rd) with V (−x) = V (x).
PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H)
unbroken PT-symmetry :⇔ H and PT have the same eigenstates ⇔ σ(H) ⊂ R
Here we assume that H has purely discrete spectrum.
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PT - symmetry[H,PT ] = 0
(Pψ)(x) := ψ(−x)
(Tψ)(x) := ψ(x)
We have in mind H = −∆+ V on L2(Rd) with V (−x) = V (x).
PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H) *
unbroken PT-symmetry :⇔ H and PT have the same eigenstates ⇔ σ(H) ⊂ R
Here we assume that H has purely discrete spectrum.
perturbation-theory insight
H0 + ε V =: H spectrum moves at most by ε ‖V ‖ր տ
self-adjoint PT-symmetric∗
=⇒ simple eigenvalues remain real for small ε
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PT - symmetry[H,PT ] = 0
(Pψ)(x) := ψ(−x)
(Tψ)(x) := ψ(x)
We have in mind H = −∆+ V on L2(Rd) with V (−x) = V (x).
PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H) *
unbroken PT-symmetry :⇔ H and PT have the same eigenstates ⇔ σ(H) ⊂ R
Here we assume that H has purely discrete spectrum.
perturbation-theory insight
H0 + ε V =: H spectrum moves at most by ε ‖V ‖ր տ
self-adjoint PT-symmetric∗
=⇒ simple eigenvalues remain real for small ε
Moreover, let the eigenstates of H form a Riesz basis. Hψn = Enψn, H∗φn = Enφn
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PT - symmetry[H,PT ] = 0
(Pψ)(x) := ψ(−x)
(Tψ)(x) := ψ(x)
We have in mind H = −∆+ V on L2(Rd) with V (−x) = V (x).
PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H) *
unbroken PT-symmetry :⇔ H and PT have the same eigenstates ⇔ σ(H) ⊂ R
Here we assume that H has purely discrete spectrum.
perturbation-theory insight
H0 + ε V =: H spectrum moves at most by ε ‖V ‖ր տ
self-adjoint PT-symmetric∗
=⇒ simple eigenvalues remain real for small ε
Moreover, let the eigenstates of H form a Riesz basis. Hψn = Enψn, H∗φn = Enφn
=⇒ H∗ = ΘHΘ−1 where Θ :=∑
n φn〈φn, ·〉 is positive, bounded, boundedly invertible
=⇒ H is self-adjoint in(L2, 〈·,Θ ·〉
), i.e. Θ1/2HΘ−1/2 is self-adjoint in
(
L2, 〈·, ·〉)
metric
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PT - symmetry[H,PT ] = 0
(Pψ)(x) := ψ(−x)
(Tψ)(x) := ψ(x)
We have in mind H = −∆+ V on L2(Rd) with V (−x) = V (x).
PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H) *
unbroken PT-symmetry :⇔ H and PT have the same eigenstates ⇔ σ(H) ⊂ R
Here we assume that H has purely discrete spectrum.
perturbation-theory insight
H0 + ε V =: H spectrum moves at most by ε ‖V ‖ր տ
self-adjoint PT-symmetric∗
=⇒ simple eigenvalues remain real for small ε
Moreover, let the eigenstates of H form a Riesz basis. Hψn = Enψn, H∗φn = Enφn
=⇒ H∗ = ΘHΘ−1 where Θ :=∑
n φn〈φn, ·〉 is positive, bounded, boundedly invertible
=⇒ H is self-adjoint in(L2, 〈·,Θ ·〉
), i.e. Θ1/2HΘ−1/2 is self-adjoint in
(
L2, 〈·, ·〉)
metric
Albeverio-Fei-Kurasov, Bender-Brody-Jones, Caliceti-Graffi-Sjostrand, Fring, Graefe-Schubert,
Kretschmer-Szymanowski, Langer-Tretter, Mostafazadeh, Scholtz-Geyer-Hahne, Znojil, . . .
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Speculations about “unbounded metric”[Kretschmer, Szymanowski 2004], [Mostafazadeh 2012], [Bender, Kuzhel 2012]
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Speculations about “unbounded metric”[Kretschmer, Szymanowski 2004], [Mostafazadeh 2012], [Bender, Kuzhel 2012]
H :=
1 2
0 1
on C2 satisfies H∗Θ = ΘH with Θ :=
0 0
0 1
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Speculations about “unbounded metric”[Kretschmer, Szymanowski 2004], [Mostafazadeh 2012], [Bender, Kuzhel 2012]
H :=
1 2
0 1
on C2 satisfies H∗Θ = ΘH with Θ :=
0 0
0 1
↓ Mostafazadeh’s construction
h = 1 on C
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Speculations about “unbounded metric”[Kretschmer, Szymanowski 2004], [Mostafazadeh 2012], [Bender, Kuzhel 2012]
H :=
1 2
0 1
on C2 satisfies H∗Θ = ΘH with Θ :=
0 0
0 1
↓ Mostafazadeh’s construction
h = 1 on C
In ∞-dimensional spaces: ¡ similar examples with Θ > 0 invertible but Θ−1 unbounded !
¡ possible 〈φn, ψn〉 6= 0 for all n but 〈φn, ψn〉 −−−−→n→∞
0 !
¡ unbounded Θ or Θ−1 always exist !
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Speculations about “unbounded metric”[Kretschmer, Szymanowski 2004], [Mostafazadeh 2012], [Bender, Kuzhel 2012]
H :=
1 2
0 1
on C2 satisfies H∗Θ = ΘH with Θ :=
0 0
0 1
↓ Mostafazadeh’s construction
h = 1 on C
In ∞-dimensional spaces: ¡ similar examples with Θ > 0 invertible but Θ−1 unbounded !
¡ possible 〈φn, ψn〉 6= 0 for all n but 〈φn, ψn〉 −−−−→n→∞
0 !
¡ unbounded Θ or Θ−1 always exist !
Moreover: ¡ physically relevant quantities are not preserved !
(continuous spectrum, pseudospectrum)
¡ spectral instablities !
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Speculations about “unbounded metric”[Kretschmer, Szymanowski 2004], [Mostafazadeh 2012], [Bender, Kuzhel 2012]
H :=
1 2
0 1
on C2 satisfies H∗Θ = ΘH with Θ :=
0 0
0 1
↓ Mostafazadeh’s construction
h = 1 on C
In ∞-dimensional spaces: ¡ similar examples with Θ > 0 invertible but Θ−1 unbounded !
¡ possible 〈φn, ψn〉 6= 0 for all n but 〈φn, ψn〉 −−−−→n→∞
0 !
¡ unbounded Θ or Θ−1 always exist !
Moreover: ¡ physically relevant quantities are not preserved !
(continuous spectrum, pseudospectrum)
¡ spectral instablities !
concept of “unbounded metric” is trivial and physically doubtful
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Mathematical frameworksto understand PTH PT = H in a more general setting
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Mathematical frameworksto understand PTH PT = H in a more general setting than:
• H = −∆+ V on L2(Rd) with V (−x) = V (x)
• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)
Remark. In general, a PT-symmetric operator is not similarto a self-adjoint, normal or spectral operator.
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Mathematical frameworksto understand PTH PT = H in a more general setting than:
• H = −∆+ V on L2(Rd) with V (−x) = V (x)
• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)
Remark. In general, a PT-symmetric operator is not similarto a self-adjoint, normal or spectral operator.
1. antilinear symmetry [H, S] = 0 with S antiunitary (bijective and 〈Sφ, Sψ〉 = 〈ψ, φ〉)
e.g. S := PT
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Mathematical frameworksto understand PTH PT = H in a more general setting than:
• H = −∆+ V on L2(Rd) with V (−x) = V (x)
• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)
Remark. In general, a PT-symmetric operator is not similarto a self-adjoint, normal or spectral operator.
1. antilinear symmetry [H, S] = 0 with S antiunitary (bijective and 〈Sφ, Sψ〉 = 〈ψ, φ〉)
e.g. S := PT
2. self-adjointness in Krein spaces H is self-adjoint in an indefinite inner product space
e.g. [·, ·] := 〈·,P·〉 after noticing PHP = THT = H∗ [Langer, Tretter 2004]
![Page 34: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/34.jpg)
Mathematical frameworksto understand PTH PT = H in a more general setting than:
• H = −∆+ V on L2(Rd) with V (−x) = V (x)
• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)
Remark. In general, a PT-symmetric operator is not similarto a self-adjoint, normal or spectral operator.
1. antilinear symmetry [H, S] = 0 with S antiunitary (bijective and 〈Sφ, Sψ〉 = 〈ψ, φ〉)
e.g. S := PT
2. self-adjointness in Krein spaces H is self-adjoint in an indefinite inner product space
e.g. [·, ·] := 〈·,P·〉 after noticing PHP = THT = H∗ [Langer, Tretter 2004]
3. J-self-adjointness H∗ = JHJ with J conjugation (involutive and 〈Jφ, Jψ〉 = 〈ψ, φ〉)
e.g. J := T after noticing THT = PHP = H∗ [Borisov, D.K. 2007]
![Page 35: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/35.jpg)
Mathematical frameworksto understand PTH PT = H in a more general setting than:
• H = −∆+ V on L2(Rd) with V (−x) = V (x)
• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)
Remark. In general, a PT-symmetric operator is not similarto a self-adjoint, normal or spectral operator.
1. antilinear symmetry [H, S] = 0 with S antiunitary (bijective and 〈Sφ, Sψ〉 = 〈ψ, φ〉)
e.g. S := PT
2. self-adjointness in Krein spaces H is self-adjoint in an indefinite inner product space
e.g. [·, ·] := 〈·,P·〉 after noticing PHP = THT = H∗ [Langer, Tretter 2004]
3. J-self-adjointness H∗ = JHJ with J conjugation (involutive and 〈Jφ, Jψ〉 = 〈ψ, φ〉)
e.g. J := T after noticing THT = PHP = H∗ [Borisov, D.K. 2007]
Remark. In general (in ∞-dimensional spaces), all the classes of operators are unrelated.[Siegl 2008]
![Page 36: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/36.jpg)
¿ Physical relevance ?
![Page 37: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/37.jpg)
¿ Physical relevance ?suggestions:
• nuclear physics [Scholtz, Geyer, Hahne 1992]
• optics [Klaiman, Gunther, Moiseyev 2008], [Schomerus 2010], [West, Kottos, Prosen 2010]
• solid state physics [Bendix, Fleischmann, Kottos, Shapiro 2009]
• superconductivity [Rubinstein, Sternberg, Ma 2007]
• electromagnetism [Ruschhaupt, Delgado, Muga 2005], [Mostafazadeh 2009]
experiments:
• optics [Guo et al. 2009], [Longhi 2009], [Ruter et al. 2010]
• mechanics [Bender, Berntson, Parker, Samuel 2012]
![Page 38: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/38.jpg)
¿ Physical relevance ?suggestions:
• nuclear physics [Scholtz, Geyer, Hahne 1992]
• optics [Klaiman, Gunther, Moiseyev 2008], [Schomerus 2010], [West, Kottos, Prosen 2010]
• solid state physics [Bendix, Fleischmann, Kottos, Shapiro 2009]
• superconductivity [Rubinstein, Sternberg, Ma 2007]
• electromagnetism [Ruschhaupt, Delgado, Muga 2005], [Mostafazadeh 2009]
experiments:
• optics [Guo et al. 2009], [Longhi 2009], [Ruter et al. 2010]
• mechanics [Bender, Berntson, Parker, Samuel 2012]
¡ but !
“So far, there have been no experiments that prove clearly and definitively that quantum
systems defined by non-Hermitian PT-symmetric Hamiltonians do exist in nature.”
[Bender 2007]
![Page 39: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/39.jpg)
The simplest PT - symmetric model[D.K., Bıla, Znojil 2006]
H := L2(−π2 ,
π2 )
Hαψ := −ψ′′ , D(Hα) :=ψ ∈W 2,2(−π
2 ,π2 )
∣∣∣ ψ′(±π2 ) + iαψ(±π
2 ) = 0, α ∈ R
-D
0 1 2 3 4 5 6
![Page 40: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/40.jpg)
The simplest PT - symmetric model[D.K., Bıla, Znojil 2006]
H := L2(−π2 ,
π2 )
Hαψ := −ψ′′ , D(Hα) :=ψ ∈W 2,2(−π
2 ,π2 )
∣∣∣ ψ′(±π2 ) + iαψ(±π
2 ) = 0, α ∈ R
-D
Theorem 1. Hα is an m-sectorial operator with compact resolvent satisfying
H∗α = H−α = THαT (T-self-adjointness)
0 1 2 3 4 5 6
![Page 41: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/41.jpg)
The simplest PT - symmetric model[D.K., Bıla, Znojil 2006]
H := L2(−π2 ,
π2 )
Hαψ := −ψ′′ , D(Hα) :=ψ ∈W 2,2(−π
2 ,π2 )
∣∣∣ ψ′(±π2 ) + iαψ(±π
2 ) = 0, α ∈ R
-D
Theorem 1. Hα is an m-sectorial operator with compact resolvent satisfying
H∗α = H−α = THαT (T-self-adjointness)
Theorem 2. σ(Hα) = α2 ∪ n2∞n=1
0 1 2 3 4 5 6alpha
5
10
15
20
25
![Page 42: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/42.jpg)
The simplest PT - symmetric model[D.K., Bıla, Znojil 2006]
H := L2(−π2 ,
π2 )
Hαψ := −ψ′′ , D(Hα) :=ψ ∈W 2,2(−π
2 ,π2 )
∣∣∣ ψ′(±π2 ) + iαψ(±π
2 ) = 0, α ∈ R
-D
Theorem 1. Hα is an m-sectorial operator with compact resolvent satisfying
H∗α = H−α = THαT (T-self-adjointness)
Theorem 2. σ(Hα) = α2 ∪ n2∞n=1
0 1 2 3 4 5 6alpha
5
10
15
20
25
Corollary. The spectrum of Hα is
always real,
simple if α 6∈ Z \ 0.
![Page 43: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/43.jpg)
Scattering realisation in QM[Hernandez-Coronado, D.K., Siegl 2011]
scattering by a compactly supported even potential V : −ψ′′ + V ψ = k2ψ k > 0
x0 π
ψin(x) = eikx +Re−ikx ψout(x) = T eikx¿ ψ(x) ?
![Page 44: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/44.jpg)
Scattering realisation in QM[Hernandez-Coronado, D.K., Siegl 2011]
scattering by a compactly supported even potential V : −ψ′′ + V ψ = k2ψ k > 0
x0 π
ψin(x) = eikx +Re−ikx ψout(x) = T eikx¿ ψ(x) ?
perfect transmission =⇒ non-linear
(i.e. R = 0)
−ψ′′ + V ψ = k2ψ in (0, π)
ψ′ − ikψ = 0 at 0, π
![Page 45: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/45.jpg)
Scattering realisation in QM[Hernandez-Coronado, D.K., Siegl 2011]
scattering by a compactly supported even potential V : −ψ′′ + V ψ = k2ψ k > 0
x0 π
ψin(x) = eikx +Re−ikx ψout(x) = T eikx¿ ψ(x) ?
perfect transmission =⇒ non-linear
(i.e. R = 0)
−ψ′′ + V ψ = k2ψ in (0, π)
ψ′ − ikψ = 0 at 0, π
solutions given by a non-self-adjoint PT-symmetric spectral problem:
−ψ′′ + V ψ = µ(α)ψ in (0, π)
ψ′ + iαψ = 0 at 0, π
µ(α) = α2
↓
non-Hermitian
![Page 46: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/46.jpg)
The metric operator[D.K., Siegl, Zelezny 2011]
Theorem 3. Let α 6∈ Z \ 0.
Then Hα is similar to a self-adjoint operator hα := ΩHα Ω−1 with the metric
Θ := Ω∗Ω = I +K
K(x, y) = 2 iπ ei
α
2(x−y) sin
(α2 (x− y)
)+ iα
π
(|y − x| − π
)sgn(y − x) + α2
π
(π2
4 − xy −π2 |y − x|
)
![Page 47: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/47.jpg)
The metric operator[D.K., Siegl, Zelezny 2011]
Theorem 3. Let α 6∈ Z \ 0.
Then Hα is similar to a self-adjoint operator hα := ΩHα Ω−1 with the metric
Θ := Ω∗Ω = I +K
K(x, y) = 2 iπ ei
α
2(x−y) sin
(α2 (x− y)
)+ iα
π
(|y − x| − π
)sgn(y − x) + α2
π
(π2
4 − xy −π2 |y − x|
)
Moreover, hα = −∆N + α2 χN0 〈χ
N0 , ·〉 !!! χN
0 (x) := π−1/2 (Neumann ground state)
non-Hermitian Hα ←→ non-local hα
![Page 48: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/48.jpg)
The metric operator[D.K., Siegl, Zelezny 2011]
Theorem 3. Let α 6∈ Z \ 0.
Then Hα is similar to a self-adjoint operator hα := ΩHα Ω−1 with the metric
Θ := Ω∗Ω = I +K
K(x, y) = 2 iπ ei
α
2(x−y) sin
(α2 (x− y)
)+ iα
π
(|y − x| − π
)sgn(y − x) + α2
π
(π2
4 − xy −π2 |y − x|
)
Moreover, hα = −∆N + α2 χN0 〈χ
N0 , ·〉 !!! χN
0 (x) := π−1/2 (Neumann ground state)
non-Hermitian Hα ←→ non-local hα
Proof. “backward usage of the spectral theorem” [D.K. 2008]
En =
α2 if n = 0
n2 = ENn = ED
n if n ≥ 1φn(x) =
1√πeiα(x+
π
2) if n = 0
χNn (x) + i α
n χDn (x) if n ≥ 1
Ω :=
∞∑
n=0
χNn 〈φn, ·〉 = χN
0 〈φ0, ·〉 − χN0 〈χ
N0 , ·〉+
∞∑
n=0
χNn 〈χ
Nn , ·〉 − iα
∞∑
n=1
1
nχNn 〈χ
Dn , ·〉
= χN0 〈φ0, ·〉 − χ
N0 〈χ
N0 , ·〉+ I + αp (−∆D)−1 (−∆D = p∗p) q
.e.d.
![Page 49: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/49.jpg)
The C operator
Definition ([Bender, Brody Jones 2002], [Albeverio, Kuzhel 2005]).
Let H be P-self-adjoint and C bounded.
H is C-symmetric :⇐⇒
[H,C] = 0
C2 = I
PC is a metric for H
![Page 50: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/50.jpg)
The C operator
Definition ([Bender, Brody Jones 2002], [Albeverio, Kuzhel 2005]).
Let H be P-self-adjoint and C bounded.
H is C-symmetric :⇐⇒
[H,C] = 0
C2 = I
PC is a metric for H
Thus: C := PΘ for Θ satisfying (PΘ)2 = I
![Page 51: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/51.jpg)
The C operator
Definition ([Bender, Brody Jones 2002], [Albeverio, Kuzhel 2005]).
Let H be P-self-adjoint and C bounded.
H is C-symmetric :⇐⇒
[H,C] = 0
C2 = I
PC is a metric for H
Thus: C := PΘ for Θ satisfying (PΘ)2 = I
Our explicit result:
C = P+ L with
L(x, y) = α e−iα(y+x)[tan(απ
2 )− i sgn(y + x)]
(|α| < 1)
![Page 52: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/52.jpg)
General PT - symmetric case[D.K., Siegl 2010]
Hα,βψ := −ψ′′ , D(Hα,β) :=ψ ∈W 2,2(−π
2 ,π2 )
∣∣∣ ψ′(±π2 ) + (iα± β)ψ(±π
2 ) = 0
β > 0 β = 0 β < 0
![Page 53: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/53.jpg)
General PT - symmetric case[D.K., Siegl 2010]
Hα,βψ := −ψ′′ , D(Hα,β) :=ψ ∈W 2,2(−π
2 ,π2 )
∣∣∣ ψ′(±π2 ) + (iα± β)ψ(±π
2 ) = 0
β > 0 β = 0 β < 0
Θ = I +K with
K(x, y) = eiα(x−y)−β|x−y| [c+ i α sgn(x− y)]
c ∈ R
Θ > 0 if β > 0 large or c2 + α2 small
![Page 54: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/54.jpg)
Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R)
![Page 55: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/55.jpg)
Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R), D(H) :=
ψ ∈ L2(R) | Hψ ∈ L2(R)
• H is m-accretive ⇒ ℜσ(H) ≥ 0 [Edmunds, Evans 1987]
• H has purely discrete spectrum [Caliceti, Graffi, Maioli 1980], [Mezinescu 2001]
• all eigenvalues of H are real [Dorey, Dunning, Tateo 2001], [Shin 2002]
![Page 56: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/56.jpg)
Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R), D(H) :=
ψ ∈ L2(R) | Hψ ∈ L2(R)
• H is m-accretive ⇒ ℜσ(H) ≥ 0 [Edmunds, Evans 1987]
• H has purely discrete spectrum [Caliceti, Graffi, Maioli 1980], [Mezinescu 2001]
• all eigenvalues of H are real [Dorey, Dunning, Tateo 2001], [Shin 2002]
¿ Does H possess metric ?
![Page 57: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/57.jpg)
Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R), D(H) :=
ψ ∈ L2(R) | Hψ ∈ L2(R)
• H is m-accretive ⇒ ℜσ(H) ≥ 0 [Edmunds, Evans 1987]
• H has purely discrete spectrum [Caliceti, Graffi, Maioli 1980], [Mezinescu 2001]
• all eigenvalues of H are real [Dorey, Dunning, Tateo 2001], [Shin 2002]
¿ Does H possess metric ?
• eigenfunctions of H form a complete set
![Page 58: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/58.jpg)
Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R), D(H) :=
ψ ∈ L2(R) | Hψ ∈ L2(R)
• H is m-accretive ⇒ ℜσ(H) ≥ 0 [Edmunds, Evans 1987]
• H has purely discrete spectrum [Caliceti, Graffi, Maioli 1980], [Mezinescu 2001]
• all eigenvalues of H are real [Dorey, Dunning, Tateo 2001], [Shin 2002]
¿ Does H possess metric ?
• eigenfunctions of H form a complete set
Theorem. Metric operator for H does not exist.
![Page 59: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/59.jpg)
Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R), D(H) :=
ψ ∈ L2(R) | Hψ ∈ L2(R)
• H is m-accretive ⇒ ℜσ(H) ≥ 0 [Edmunds, Evans 1987]
• H has purely discrete spectrum [Caliceti, Graffi, Maioli 1980], [Mezinescu 2001]
• all eigenvalues of H are real [Dorey, Dunning, Tateo 2001], [Shin 2002]
¿ Does H possess metric ?
• eigenfunctions of H form a complete set
Theorem. Metric operator for H does not exist.
Proof.
• Let metric exist ⇒∥∥(H − z)−1
∥∥ ≤ C
|ℑz|, ∀z ∈ C, ℑz 6= 0
![Page 60: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/60.jpg)
Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R), D(H) :=
ψ ∈ L2(R) | Hψ ∈ L2(R)
• H is m-accretive ⇒ ℜσ(H) ≥ 0 [Edmunds, Evans 1987]
• H has purely discrete spectrum [Caliceti, Graffi, Maioli 1980], [Mezinescu 2001]
• all eigenvalues of H are real [Dorey, Dunning, Tateo 2001], [Shin 2002]
¿ Does H possess metric ?
• eigenfunctions of H form a complete set
Theorem. Metric operator for H does not exist.
Proof.
• Let metric exist ⇒∥∥(H − z)−1
∥∥ ≤ C
|ℑz|, ∀z ∈ C, ℑz 6= 0
•∥∥(H − σz)−1
∥∥ = σ−1∥∥(Hh − z)
−1∥∥ with Hh := −h2
d2
dx2+ ix3, h := σ−5/6
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Imaginary cubic oscillator[D.K., Siegl 2012]
H = −d2
dx2+ ix3 on L2(R), D(H) :=
ψ ∈ L2(R) | Hψ ∈ L2(R)
• H is m-accretive ⇒ ℜσ(H) ≥ 0 [Edmunds, Evans 1987]
• H has purely discrete spectrum [Caliceti, Graffi, Maioli 1980], [Mezinescu 2001]
• all eigenvalues of H are real [Dorey, Dunning, Tateo 2001], [Shin 2002]
¿ Does H possess metric ?
• eigenfunctions of H form a complete set
Theorem. Metric operator for H does not exist.
Proof.
• Let metric exist ⇒∥∥(H − z)−1
∥∥ ≤ C
|ℑz|, ∀z ∈ C, ℑz 6= 0
•∥∥(H − σz)−1
∥∥ = σ−1∥∥(Hh − z)
−1∥∥ with Hh := −h2
d2
dx2+ ix3, h := σ−5/6
•∥∥(Hh − z)
−1∥∥ = O(h−n), ∀n > 0 ⇒ contradiction q.e.d.
[Davies 1999]
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Pseudospectra and PT - symmetry[work in progress with Siegl and Tater]
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Pseudospectra and PT - symmetry[work in progress with Siegl and Tater]
σε(H) :=z ∈ C
∣∣ ‖(H − z)−1‖ > ε−1
[Trefethen, Embree 2005], [Davies 2007]
![Page 64: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/64.jpg)
Pseudospectra and PT - symmetry[work in progress with Siegl and Tater]
σε(H) :=z ∈ C
∣∣ ‖(H − z)−1‖ > ε−1
[Trefethen, Embree 2005], [Davies 2007]
• H is self-adjoint =⇒ ‖(H − z)−1‖ =1
dist(z, σ(H)
) =⇒ trivial pseudospectrum
![Page 65: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/65.jpg)
Pseudospectra and PT - symmetry[work in progress with Siegl and Tater]
σε(H) :=z ∈ C
∣∣ ‖(H − z)−1‖ > ε−1
[Trefethen, Embree 2005], [Davies 2007]
• H is self-adjoint =⇒ ‖(H − z)−1‖ =1
dist(z, σ(H)
) =⇒ trivial pseudospectrum
• For non-self-adjoint operators, pseudospectra more relevant than spectra !
![Page 66: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/66.jpg)
Pseudospectra and PT - symmetry[work in progress with Siegl and Tater]
σε(H) :=z ∈ C
∣∣ ‖(H − z)−1‖ > ε−1
[Trefethen, Embree 2005], [Davies 2007]
• H is self-adjoint =⇒ ‖(H − z)−1‖ =1
dist(z, σ(H)
) =⇒ trivial pseudospectrum
• For non-self-adjoint operators, pseudospectra more relevant than spectra !
• σε(H) =⋃
‖V ‖<ε
σ(H + V ) =⇒ spectral instabilities
![Page 67: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/67.jpg)
Pseudospectra and PT - symmetry[work in progress with Siegl and Tater]
σε(H) :=z ∈ C
∣∣ ‖(H − z)−1‖ > ε−1
[Trefethen, Embree 2005], [Davies 2007]
• H is self-adjoint =⇒ ‖(H − z)−1‖ =1
dist(z, σ(H)
) =⇒ trivial pseudospectrum
• For non-self-adjoint operators, pseudospectra more relevant than spectra !
• σε(H) =⋃
‖V ‖<ε
σ(H + V ) =⇒ spectral instabilities
• ∃ metric =⇒ trivial pseudospectrum
ℜ (z)
ℑ(z
)
0 5 10 15 20 25 30 35 40 45 50−40
−30
−20
−10
0
10
20
30
40
0 5 10 15 20 25 30 35 40 45 50−40
−30
−20
−10
0
10
20
30
40
ix3 |x|3
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Conclusions
Ad PT-symmetry :
→ no extension of QM
→ rather an alternative (quasi-Hermitian) representation
→ overlooked for over 70 years
¡ some rigorous treatments still missing !
![Page 69: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/69.jpg)
Conclusions
Ad PT-symmetry :
→ no extension of QM
→ rather an alternative (quasi-Hermitian) representation
→ overlooked for over 70 years
¡ some rigorous treatments still missing !
Ad our model :
→ shamefully simple
→ closed fomulae for the spectrum, metric operator, self-adjoint counterpart, etc.
→ rigorous treatment
¡ physical relevance !
![Page 70: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/70.jpg)
Conclusions
Ad PT-symmetry :
→ no extension of QM
→ rather an alternative (quasi-Hermitian) representation
→ overlooked for over 70 years
¡ some rigorous treatments still missing !
Ad our model :
→ shamefully simple
→ closed fomulae for the spectrum, metric operator, self-adjoint counterpart, etc.
→ rigorous treatment
¡ physical relevance !
Ad ix3 :
¡ metric operator does not exist !
(bad basicity properties, non-trivial pseudospectrum, spectral instabilities)
![Page 71: Nuclear Physics Institute ASCR Reˇz, Czech Republicˇphhqp11.in2p3.fr/Thursday_30_files/KrejcirikSL12.pdf · Non-Hermitian operators in quantum physics David KREJCIˇ Rˇ´IK ...](https://reader036.fdocuments.in/reader036/viewer/2022071006/5fc3c171c10327593b58742e/html5/thumbnails/71.jpg)
Collection of open problems
Some of the open problems also available in Integral Equations Operator Theory.
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My PT - symmetric life
http://gemma.ujf.cas.cz/ david/
• D.K., H. Bıla, M. Znojil: Closed formula for the metric in the Hilbert space of a PT-symmetric model ;
J. Phys. A 39 (2006), 10143–10153.
• D.K.: Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem;
J. Phys. A: Math. Theor. 41 (2008) 244012.
• D. Borisov, D.K.: PT-symmetric waveguides;
Integral Equations Operator Theory 62 (2008), 489–515.
• D.K., M. Tater: Non-Hermitian spectral effects in a PT-symmetric waveguide;
J. Phys. A: Math. Theor. 41 (2008) 244013.
• D.K., P. Siegl: PT-symmetric models in curved manifolds;
J. Phys. A: Math. Theor. 43 (2010) 485204.
• H. Hernandez-Coronado, D.K., P. Siegl: Perfect transmission scattering as a PT-symmetric spectral problem;
Phys. Lett. A 375 (2011), 2149–2152.
• D. Borisov, D.K.: The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions;
Asympt. Anal. 76 (2012), 49–59.
• D.K., P. Siegl, J. Zelezny: Non-Hermitian PT-symmetric Sturm-Liouville operators and to them similar Hamiltonians;
arXiv:1108.4946 [math.SP] (2011).
• D. Kochan, D.K., R. Novak, P. Siegl: The Pauli equation with complex boundary conditions;
J. Phys. A: Math. Theor., to appear; arXiv:1203.5011 [quant.ph] (2012).
• P. Siegl, D.K.: Metric operator for the imaginary cubic oscillator does not exist;
arXiv:1208.1866 [quant-ph] (2012).