nuclear physics institute ascr reˇz, czech...

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.

Hors de ligne

(Outline)

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

¿ QM with non-Hermitian operators ?

R I

H∗ = H HPT = H

C

Imaginary Numbers by Yves Tanguy, 1954

(Museo Thyssen-Bornemisza, Madrid)

Insignificant non-Hermiticity

Example 1. evolution operator U(t) = exp(−itH) :

iU(t) = H U(t)

U(0) = I



iU(t) = H U(t)

U(0) = I

Example 2. resolvent operator R(z) = (H − z)−1, z ∈ C



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 .

Technical non-Hermiticity

X X XX XX

X X XX XX


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], . . .



θ = 0 ℑθ > 0

X X XX XX

X X XX XX


Example 2. adiabatic transition probability for H(t) := ~γ(t/τ) · ~σ, τ →∞

[Berry 1990], [Joye, Kunz, Pfister 1991], [Jaksic, Segert 1993], . . .



θ = 0 ℑθ > 0

X X XX XX

X X XX XX


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], . . .

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, . . .

¿ 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.




¡ 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

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 ?

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).


(Pψ)(x) := ψ(−x)

(Tψ)(x) := ψ(x)


PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H)


(Pψ)(x) := ψ(−x)

(Tψ)(x) := ψ(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.


(Pψ)(x) := ψ(−x)

(Tψ)(x) := ψ(x)


PT is an antilinear symmetry =⇒ in general only: λ ∈ σ(H)⇔ λ ∈ σ(H) *



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


(Pψ)(x) := ψ(−x)

(Tψ)(x) := ψ(x)









Moreover, let the eigenstates of H form a Riesz basis. Hψn = Enψn, H∗φn = Enφn


(Pψ)(x) := ψ(−x)

(Tψ)(x) := ψ(x)










=⇒ 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


(Pψ)(x) := ψ(−x)

(Tψ)(x) := ψ(x)










=⇒ 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, . . .

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


H :=

1 2

0 1


0 0

0 1

↓ Mostafazadeh’s construction

h = 1 on C


H :=

1 2

0 1


0 0

0 1


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 !


H :=

1 2

0 1


0 0

0 1


h = 1 on C



0 !


Moreover: ¡ physically relevant quantities are not preserved !

(continuous spectrum, pseudospectrum)

¡ spectral instablities !


H :=

1 2

0 1


0 0

0 1


h = 1 on C



0 !


Moreover: ¡ physically relevant quantities are not preserved !

(continuous spectrum, pseudospectrum)

¡ spectral instablities !

concept of “unbounded metric” is trivial and physically doubtful

Mathematical frameworksto understand PTH PT = H in a more general setting

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.



• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)


1. antilinear symmetry [H, S] = 0 with S antiunitary (bijective and 〈Sφ, Sψ〉 = 〈ψ, φ〉)

e.g. S := PT



• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)



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]



• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)



e.g. S := PT



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]



• (Pψ)(x) := ψ(−x), (Tψ)(x) := ψ(x)



e.g. S := PT



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]

¿ Physical relevance ?

¿ 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]

¿ 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]

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


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


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 2. σ(Hα) = α2 ∪ n2∞n=1

0 1 2 3 4 5 6alpha

5

10

15

20

25


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 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.

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) ?



x0 π


perfect transmission =⇒ non-linear

(i.e. R = 0)

−ψ′′ + V ψ = k2ψ in (0, π)

ψ′ − ikψ = 0 at 0, π



x0 π


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

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|

)




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




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.

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

The C operator




[H,C] = 0

C2 = I


Thus: C := PΘ for Θ satisfying (PΘ)2 = I

The C operator




[H,C] = 0

C2 = I


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)

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

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

Imaginary cubic oscillator[D.K., Siegl 2012]

H = −d2

dx2+ ix3 on L2(R)


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]


H = −d2

dx2+ ix3 on L2(R), D(H) :=

ψ ∈ L2(R) | Hψ ∈ L2(R)




¿ Does H possess metric ?


H = −d2

dx2+ ix3 on L2(R), D(H) :=

ψ ∈ L2(R) | Hψ ∈ L2(R)





• eigenfunctions of H form a complete set


H = −d2

dx2+ ix3 on L2(R), D(H) :=

ψ ∈ L2(R) | Hψ ∈ L2(R)






Theorem. Metric operator for H does not exist.


H = −d2

dx2+ ix3 on L2(R), D(H) :=

ψ ∈ L2(R) | Hψ ∈ L2(R)







Proof.

• Let metric exist ⇒∥∥(H − z)−1

∥∥ ≤ C

|ℑz|, ∀z ∈ C, ℑz 6= 0


H = −d2

dx2+ ix3 on L2(R), D(H) :=

ψ ∈ L2(R) | Hψ ∈ L2(R)







Proof.


∥∥ ≤ 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


H = −d2

dx2+ ix3 on L2(R), D(H) :=

ψ ∈ L2(R) | Hψ ∈ L2(R)







Proof.


∥∥ ≤ 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]

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]


σε(H) :=z ∈ C

∣∣ ‖(H − z)−1‖ > ε−1


• H is self-adjoint =⇒ ‖(H − z)−1‖ =1

dist(z, σ(H)

) =⇒ trivial pseudospectrum


σε(H) :=z ∈ C

∣∣ ‖(H − z)−1‖ > ε−1



dist(z, σ(H)


• For non-self-adjoint operators, pseudospectra more relevant than spectra !


σε(H) :=z ∈ C

∣∣ ‖(H − z)−1‖ > ε−1



dist(z, σ(H)



• σε(H) =⋃

‖V ‖<ε

σ(H + V ) =⇒ spectral instabilities


σε(H) :=z ∈ C

∣∣ ‖(H − z)−1‖ > ε−1



dist(z, σ(H)



• 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

Conclusions

Ad PT-symmetry :

→ no extension of QM

→ rather an alternative (quasi-Hermitian) representation

→ overlooked for over 70 years

¡ some rigorous treatments still missing !

Conclusions

Ad PT-symmetry :





Ad our model :

→ shamefully simple

→ closed fomulae for the spectrum, metric operator, self-adjoint counterpart, etc.

→ rigorous treatment

¡ physical relevance !

Conclusions

Ad PT-symmetry :





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)

Collection of open problems

Some of the open problems also available in Integral Equations Operator Theory.

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;


• D.K., P. Siegl: PT-symmetric models in curved manifolds;


• 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).

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