Spectral analysis of some non-self-adjoint operatorsgemma.ujf.cas.cz/~siegl/Data/pdf/ConfContr/Toulouse/2012_Siegl... · Spectral analysis of some non-self-adjoint operators Petr

Introduction S-L Operators Example 2D operators ix3 Conclusions

Spectral analysis of some non-self-adjointoperators

Petr Siegl

GFM University of Lisbon, Portugal

Based on:1. D. Krejcirık, P. Siegl, and J. Zelezny, On the similarity of Sturm-Liouville operators withnon-Hermitian boundary conditions to self-adjoint and normal operators, arXiv:1108.4946.2. D. Krejcirık, P. Siegl, PT -symmetric models in curved manifolds, Journal of Physics A:Mathematical and Theoretical, 2010, 43.3. P. Siegl, D. Krejcirık, P. Siegl, Metric operator for imaginary cubic oscillator does not exist,arXiv:1208.1866.


Outline

1. Introduction• PT -symmetric operators• Motivation and mathematical approach

2. Sturm-Liouville operators• Structure of similarity transformation• Examples: closed formulae

3. 2D generalizations• Strips in curved manifolds• Waveguides

4. Imaginary cubic oscillator• Spectrum• Metric operator


PT -symmetric operators

PT -symmetric spectral problems

• H = − d2

dx2 + ix3 has real and discrete spectrum [BeBo98], [DoDuTa01], [Sh02]

• the reality of spectrum due to PT -symmetry• [PT ,H ] = 0• parity P, (Pψ)(x) = ψ(−x)• time reversal T , (T ψ)(x) = ψ(x)

Simple observations

• PT -symmetry is not sufficient for real spectrum• some PT -symmetric operators are similar to self-adjoint or normal operators∃Ω,Ω−1 ∈ B(H): ΩHΩ−1 is self-adjoint or normal

[BeBo98] 1998 Bender and Boettcher, Physical Review Letters 80.[DoDuTa01] 2001 Dorey, Dunning, Tateo: Journal of Physics A: Mathematical and General 34.[Sh02] 2002 Shin, Communications in Mathematical Physics 229.


MotivationRecent applications in physics

• experimental results in optics [KlGuMo08], [RuMaGaChSeKi10], [Lo10], [Re12]

• superconductivity [RuStMa07], [RuStZu10] , solid state [BeFlKoSh08]

• electromagnetism [RuDeMu05], [Mo09], nuclear physics [ScGeHa92] , QM [HCKrSi11]

Quantum mechanics

• similarity transformation → alternative representation of s-a operators

h := ΩHΩ−1, h∗ = h

• no “extension” of QM

[BeFlKoSh08] 2008 Bendix, Fleischmann, Kottos, and Shapiro, Physical Review Letters 103,[Di61] 1961 Dieudonne, Proceedings Of The International Symposium on Linear Spaces,[HCKrSi10] 2011, Hernandez-Coronado, Krejcirık, Siegl, Physics Letters A 375,[KlGuMo08] 2008 Klaiman, Gunther, and Moiseyev, Physical Review Letters 101,[Lo10] 2010 Longhi, Physical Review Letters 105,[Mo09] 2009 Mostafazadeh, Physical Review Letters 102,[Re12] 2012 A. Regensburger et. al., Nature 488, 167,[RuStMa07] 2007 Rubinstein, Sternberg, and Ma, Physical Review Letters 99,[RuStZu10] 2010 Rubinstein, Sternberg, and Zumbrun, Archive for Rational Mechanics and Analysis 195,[RuDeMu05] 2005 Ruschhaupt, Delgado, Muga, Journal of Physics A: Mathematical and General 38,[RuMaGaChSeKi10] 2010 Ruter, Makris, El-Ganainy, Christodoulides, Segev, and Kip, Nature Physics 6,[ScGeHa92] 1992 Scholtz, Geyer, and Hahne, Annals of Physics 213.


Mathematical approachKrein spaces

• self-adjoint operators in Krein space with [·,P·] [LaTr04]

• H = PH∗P• spectrum symmetric w.r.t. the real axis• spectrum of definite type + perturbation stability

Example in L2(−1, 1) [Zn01]

• Hε = −∆D + i ε sgnx• Dom (Hε) = W 1,2

0 (−1, 1) ∩W 2,2(−1, 1)

0 2 4 6 8 10 12 14Z

10

20

30

40Re Λ

2 4 6 8 10 12 14Ε

-10

-5

5

10Im Λ

[LaTr04] 2004 Langer, Tretter, Czechoslovak Journal of Physics 54,

[Zn01] 2001 Znojil, Physics Letters A 285.


Mathematical approach

J -self-adjoint operators

• J -self-adjoint operators approach [BoKr08]

• H = JH∗J• J is an antilinear, isometric involution:

J2 = I , ∀x, y ∈ H : 〈Jx, Jy〉 = 〈y, x〉• for PT -symmetric systems: often J = T• residual spectrum of J -s-a operators is empty

Example in L2(R) [EdEv87]

• Re V bounded from below, V ∈ L2loc(R)

• H = −d2

dx2 + V (x)

• Dom (H) = ψ ∈ L2(R) : Vψ ∈ L1loc(R), −ψ′′ + Vψ ∈ L2(R)

[EdEv87] 1987 Edmund, Evans: Spectral Theory and Differential Operators,

[BoKr08] 2008 Borisov, Krejcirık, Integral Equations and Operator Theory 62


Sturm-Liouville operator

Object of interest

• 1D Sturm-Liouville operator in L2(−a, a)• differential expression

τψ := −ψ′′ + Vψ

• boundary conditions

ψ′(±a) + c±ψ(±a) = 0

• V ∈ L∞(−a, a) complex potential, c± ∈ C• c±,V are real: self-adjoint operators• c±,V are complex: J -self-adjoint operators


Basic definitions and concepts

Definition of operator H

Hψ := −ψ′′ + Vψ

Dom (H) := ψ ∈W 2,2(−a, a) : ψ′(±a) + c±ψ(±a) = 0

Basic properties of H

• the adjoint operator H∗

H∗ψ = −ψ′′ + Vψ

Dom (H∗) = ψ ∈W 2,2(−a, a) : ψ′(±a) + c±ψ(±a) = 0

• H is an m-sectorial operator associated with the sectorial form tH

tH [ψ] := ‖ψ′‖2 + c+|ψ(a)|2 − c−|ψ(−a)|2 + 〈ψ,Vψ〉

Dom (tH ) := W 1,2(−a, a)

• spectrum of H is discrete, i.e. only isolated eigenvalues with finitemultiplicities

• H forms a holomorphic family of type (B) w.r.t. the parameters c±


Properties of H

Symmetries

• H is self-adjoint: H∗ = H , iff c± ∈ R and V (x) ∈ R• H is T -self-adjoint: H∗ = T HT

• H is P-self-adjoint: H∗ = PHP, iff c− = −c+ and V (−x) = V (x)

• H is PT -symmetric: [H ,PT ] = 0, iff c− = −c+ and V (−x) = V (x)

Eigenvalues and eigenfunctions

Hψn = λnψn

H∗φn = λnφn

Theorem [Mi62], [DSIII]

Eigenfunctions of H (together with associated functions) form a Riesz basis. H isa discrete spectral operator.

[DSIII] 1971 Dunford, Schwartz, Linear Operators, Part 3, Spectral Operators,

[Mi62] 1962 Mikhajlov, Doklady Akademii Nauk SSSR 114


Riesz basis, similarity transformation, metric operator

Riesz basis

• ψnn∈N form a Riesz basis if there exists a bounded operator ρ withbounded inverse and an orthonormal basis enn∈N such that ψn = ρen .

• eigenfunctions of H form a Riesz basis iff all eigenvalues are simple

Similarity transformation

• we search for a bounded operator Ω with bounded inverse such thath := ΩHΩ−1 is self-adjoint or normal operator

• such Ω exists iff the eigenfunctions ψnn∈N of H form a Riesz basis

Metric operator

• we search for bounded positive operator Θ with bounded inverse such thatH is self-adjoint or normal w.r.t. new inner product 〈·,Θ·〉

• such Θ exists iff the eigenfunctions ψnn∈N of H form a Riesz basis


Similarity transformations, metric operators formulaeMetric operator Θ

Θ :=∞∑

n=0

cnφn〈φn , ·〉

0 < m < cn < M <∞

Similarity transformation Ω

Ω :=∞∑

n=0

√cnen〈φn , ·〉

0 < m < cn < M <∞enn∈N form an orthonormal basis

Relations between Θ, Ω, H

Θ = Ω∗Ω, h := ΩHΩ−1

∀λn ∈ R : ΘH = H∗Θ ⇔ h = h∗

∃λn /∈ R : ΘHΘ−1H∗ = H∗ΘHΘ−1 ⇔ hh∗ = h∗h


Structure of Ω and ΘTheorem [KrSiZe11]]

Let all eigenvalues of H be simple. Then• Ω = U + L,• Θ = I + K ,

where K ,L are integral (H-S) operators and U is a unitary operator.Moreover, if en := χN

n , then U = I and Ω,Ω−1,Ω∗, (Ω∗)−1 are bounded onW 1,2(−a, a) and W 2,2(−a, a).

Corollary

h := ΩHΩ−1 is a holomorphic family of type (B) w.r.t. c± and the associatedform reads

th [ψ] = ‖ψ′‖2 + 〈(L∗ψ)′, ψ′〉+ 〈ψ′, (Mψ)′〉+ 〈(L∗ψ)′, (Mψ)′〉

+ c+[(ψ(a) + (L∗ψ)(a)

)(ψ(a) + (Mψ)(a)

)]− c−

[(ψ(−a) + (L∗ψ)(−a)

)(ψ(−a) + (Mψ)(−a)

)],

Dom (th) = W 1,2(−a, a).

where Ω = I + L, Ω−1 = I + M .


Structure of Θ and Ω

Proof and remarks

• asymptotics of EVs and EFs + analytic perturbation theory• similar h is typically non-local• “preferred” basis χN

n , U = I• PT -symmetry is not needed (but provides “nice” examples)• valid for strictly regular connected BC as well (expected)• only regular BC, e.g. periodic, very different situation [GeTk09,DjMi11]

• K is not always an integral (neither compact) operator• explicitly solvable examples?

[DjMi] 2011 Djakov, Mityagin, arXiv:1106.5774,

[GeTk] 2009 Gesztesy, Tkachenko, Journal d’Analyse Mathematique 107.


PT -symmetric exampleSimplest possible example [KrBiZn06]

Hαψ = −ψ′′

Dom (Hα) = ψ ∈W 2,2(−a, a) : ψ′(±a) + iαψ(±a) = 0

Symmetries

• H∗α = H−α• PT -symmetry: HαPT = PT Hα• P-self-adjointness: Hα = PH∗αP• T -self-adjointness: Hα = T H∗αT

Eigenvalues

σ(Hα) = α2 ∪ k2n∞n=1

kn =nπ2a

0 1 2 3 4 5Α

5

10

15

20

Λ

[KrBiZn06] 2006 Krejcirık, Bıla, Znojil, Journal of Physics A: Mathematical and General 39


PT -symmetric exampleEigenfunctions

• eigenfunctions of Hα

λ0 = α2 : ψ0(x) = A0e−iα(x+a),

λn = k2n : ψn(x) = An

(χN

n (x)− iα

knχD

n (x))

• eigenfunctions of H∗α

λ0 = α2 : φ0(x) =1√

2aeiα(x+a),

λn = k2n : φn(x) = χN

n (x) + iα

knχD

n (x)

• for a = π/2:

• kn = n• χD

n (x) =√

2/π sin n(x + π/2)• χN

n (x) =√

2/π cos n(x + π/2), χN0 (x) =

√1/π

• χD,Nn are eigenfunctions of −∆D,N

• A0, An such that 〈φn , ψm〉 = δnm


Similarity transformation ΩFormulae

Ω =∞∑

n=0

χNn 〈φn , ·〉, φn = χN

n + iα

knχD

n , φ0(x) =1√

2aeiα(x+a)

Construction of Ω

• usage of functional calculus [Kr08]

Ω =∞∑

n=0

χNn 〈φn , ·〉 =

∞∑n=1

χNn 〈χN

n , ·〉 − iα

kn

∞∑n=1

χNn 〈χD

n , ·〉

+ χN0 〈φ

N0 , ·〉+ χN

0 〈χN0 , ·〉 − χ

N0 〈χ

N0 , ·〉

=∞∑

n=0

χNn 〈χN

n , ·〉+ χN0 〈φ

N0 − χ

N0 , ·〉+ αp

∞∑n=0

1k2

nχD

n 〈χDn , ·〉

= I + χN0 〈φ

N0 − χ

N0 , ·〉+ αp(−∆D)−1

• pψ := −iψ′

• ipχDn = knχN

n , ipχNn = −knχD

n

[Kr08] 2008 Krejcirık: Journal of Physics A: Mathematical and General 41.



Ω =∞∑

n=0


n + iα

knχD

n , φ0(x) =1√

2aeiα(x+a)

Construction of Ω


Ω =∞∑

n=0

χNn 〈φn , ·〉 =

∞∑n=1

χNn 〈χN

n , ·〉 − iα

kn

∞∑n=1

χNn 〈χD

n , ·〉

+ χN0 〈φ

N0 , ·〉+ χN

0 〈χN0 , ·〉 − χ

N0 〈χ

N0 , ·〉

=∞∑

n=0

χNn 〈χN

n , ·〉+ χN0 〈φ

N0 − χ

N0 , ·〉+ αp

∞∑n=0

1k2

nχD

n 〈χDn , ·〉

= I + χN0 〈φ

N0 − χ

N0 , ·〉+ αp(−∆D)−1

• pψ := −iψ′

• ipχDn = knχN


n




Ω =∞∑

n=0


n + iα

knχD

n , φ0(x) =1√

2aeiα(x+a)

Construction of Ω


Ω =∞∑

n=0

χNn 〈φn , ·〉 =

∞∑n=1

χNn 〈χN

n , ·〉 − iα

kn

∞∑n=1

χNn 〈χD

n , ·〉

+ χN0 〈φ

N0 , ·〉+ χN

0 〈χN0 , ·〉 − χ

N0 〈χ

N0 , ·〉

=∞∑

n=0

χNn 〈χN

n , ·〉+ χN0 〈φ

N0 − χ

N0 , ·〉+ αp

∞∑n=0

1k2

nχD

n 〈χDn , ·〉

= I + χN0 〈φ

N0 − χ

N0 , ·〉+ αp(−∆D)−1

• pψ := −iψ′

• ipχDn = knχN


n



Closed form of Θ, Ω, Ω−1

Theorem [KrSiZe11]

Let α 6= kn (no degeneracies in the spectrum). Then

• Ω = I + L, Ω−1 = I + M , Θ = I + K• K ,L,M integral operators with kernels

L(x, y) =iα2a

(y − a sgn (y − x)) +1

2a(

eiα(y+a) − 1)

M(x, y) =αeiα(a−x)

sin(2αa)−α

2e−iα(x−y) (cot(2αa)− isgn (y − x))

−αe−iα(x+y)

2 sin(2αa),

K(x, y) =ia

ei α2 (y−x) sin

(α

2(y − x)

)+α2

2a(

a2 − xy)

+iα2a

(y − x)

−iα2

(2− iα(y − x)) sgn(y − x).

• for (different) special choice of cn 6= 1

K(x, y) = αe−iα(y−x) (tan(αa)− isgn (y − x))



Theorem [KrSiZe11]

Any Θ for Hα has the form

Θ = JN + c0θ1 + JNθ2 + JDθ3

c0 ∈ R+, θi are integral operators with kernels:

θ1(x, y) :=ia

eiα2 (y−x) sin

(α

2(y − x)

),

θ2(x, y) :=iα2a

(y − a sgn (y − x)) ,

θ3(x, y) :=α2

2a(

a2 − xy)−

iα2a

x −iα2

(1− iα(y − x)) sgn (y − x).

and

JD :=∞∑

n=1

cnχDn 〈χD

n , ·〉, JN :=∞∑

n=0

cnχNn 〈χN

n , ·〉

Remarks

• JD,N = I if cn = 1• JD,N are metric operators for −∆D,N : [−∆D,N , JD,N ] = 0, JD,N > 0



Theorem [KrSiZe11]

Any Θ for Hα has the form

Θ = JN + c0θ1 + JNθ2 + JDθ3

c0 ∈ R+, θi are integral operators with kernels:

θ1(x, y) :=ia

eiα2 (y−x) sin

(α

2(y − x)

),

θ2(x, y) :=iα2a

(y − a sgn (y − x)) ,

θ3(x, y) :=α2

2a(

a2 − xy)−

iα2a

x −iα2

(1− iα(y − x)) sgn (y − x).

and

JD :=∞∑

n=1

cnχDn 〈χD

n , ·〉, JN :=∞∑

n=0

cnχNn 〈χN

n , ·〉

Remarks

• JD,N = I if cn = 1• JD,N are metric operators for −∆D,N : [−∆D,N , JD,N ] = 0, JD,N > 0


Similar s-a operatorTheorem [KrSiZe11]

Let α 6= kn . Then the similar self-adjoint operator h := ΩHΩ−1 has the formh = −∆N + α2〈χN

0 , ·〉χN0 .

0 1 2 3 4 5Α

5

10

15

20

Λ

Remarks

• rank one perturbation of −∆N

• multiple EVs ⇔ Θ, Ω break down (not invertible)• h is self-adjoint (without Jordan blocks) also with multiple EVs


PT -symmetric example IIOperator

Hα,βψ := −ψ′′

Dom (Hα,β) := ψ ∈W 2,2(−a, a) : ψ′(±a) + (iα± β)ψ(±a) = 0

Eigenvalues

(k2 − α2 − β2) sin(2ak)− 2βk cos(2ak) = 0.

β > 0

0 2 4 6 8Α

5

10

15

20Re Λ

β < 0

0 1 2 3 4Α

5

10

15

20Re Λ

Theorem [KrSi10]

Let α ∈ R. If β > 0 then all eigenvalues of H are simple and real. If β < 0 thenthere are at most two complex conjugated eigenvalues.[KrSi10] 2010 Krejcirık, Siegl, Journal Of Physics A: Mathematical and Theoretical 43


PT -symmetric example II

Theorem [KrSiZe11]

Let |β| be sufficiently small. Then the metric operator of H can be found as

Θ = I + K

with

K(x, y) = e[iα−β sgn (x−y)](x−y)(c + iα sgn (x − y)), c ∈ R,

Proof and remarks

• different method: “solving” ΘHα,β = H∗α,βΘ

• Θ is positive e.g. for β small• Ω not known


PT -symmetric example III

Irregular boundary conditions

• Hψ := −ψ′′

• Dom (H) : ψ ∈W 2,2(−a, a) :

ψ(a) = eiτ1ψ(−a), ψ(0+) = eiτ2ψ(0−)

ψ′(a) = e−iτ1ψ′(−a), ψ′(0+) = e−iτ2ψ′(0−).

Spectrum

• discrete if τ1 6= π/2 and τ2 6= π/2• empty if τ1 = π/2 and τ2 6= π/2• entire C if τ1 = π/2 and τ2 = π/2

Symmetries of H

• H is PT -symmetric, P-self-adjoint, T -self-adjoint


PT -symmetric example III

Irregular boundary conditions

• Hψ := −ψ′′

• Dom (H) : ψ ∈W 2,2(−a, a) :

ψ(a) = eiτ1ψ(−a), ψ(0+) = eiτ2ψ(0−)

ψ′(a) = e−iτ1ψ′(−a), ψ′(0+) = e−iτ2ψ′(0−).

Spectrum

• discrete if τ1 6= π/2 and τ2 6= π/2• empty if τ1 = π/2 and τ2 6= π/2• entire C if τ1 = π/2 and τ2 = π/2

Symmetries of H

• H is PT -symmetric, P-self-adjoint, T -self-adjoint


PT -symmetric example IV• Hψ := −ψ′′

• Dom (H) : ψ ∈W 2,2(−a, a) :

ψ(−a) = 0, ψ(0+) = eiτψ(0−)

ψ(a) = 0, ψ′(0+) = e−iτψ′(0−).

Theorem [AlKu05,Si08,KuTr11]

Let τ 6= ±π/2. Then• All eigenvalues of H are real and read (nπ/2a)2n∈N• Metric operator and similarity transformation have form

Θ = I + i sin(τ)P sgnx, Ω = cos(τ/2) + i sin(τ/2)P sgnx

• h := ΩHΩ−1 = −∆D.Let τ = ±π/2, then σ(H) = σp(H) = C.

Remarks

• K ,L not compact• algebraic construction of Θ with help of I ,P,R [KuTr11]

[AlKu05] 2005 Albeverio, Kuzhel, Journal of Physics A: Mathematical and General 38,[KuTr11] 2011 Kuzhel, Trunk, Journal of Mathematical Analysis and Applications 379,[Si08] 2008 Siegl, Journal of Physics A: Mathematical and Theoretical 41.


2D strips in curved manifolds [KrSi10]

x1

x2

-a

a

x1

x2

PT -symmetric BC

PT -symmetric BC

perio

dic

BC

periodic

BC

−l l

a

−a

Laplace-Beltrami operator

H =− |g|−1/2∂i |g|1/2gij∂j in L2((−π, π)× (−a, a),dΩ

)Dom (H) =W 2,2 + boundary conditions

dΩ =|g|1/2dx1dx2

PT -symmetric boundary conditions

∂2Ψ(x1, a) + (iα(x1) + β(x1))Ψ(x1, a) = 0∂2Ψ(x1,−a) + (iα(x1)− β(x1))Ψ(x1,−a) = 0

[KrSi10] 2010 Krejcirık, Siegl, Journal Of Physics A: Mathematical and Theoretical 43


Constant curvature and interaction

-a

a x1

x2

x2

x1

a

-a

Constant curvature and interaction α, β

• cylinder K = 0, sphere K = 1, pseudosphere K = −1• separation of variables: m-sectoriality ⇒ σ(H2D) =

⋃m∈Z σ(Hm

1D)

• all eigenvalues simple ⇒ H2D is similar to normal, s-a operator• for K = 0 full answer for spectrum (spectra of 1D operators)• positive curvature: spectrum remains real• negative curvature: complex eigenvalues• partial answer only, valid for “large” EVs


Constant curvature and interaction - numericsPositive curvature

0 2 4 6 8Α

5

10

15

20Λ

Negative curvature

0 1 2 3 4Α

5

10

15

20Re Λ

1 2 3 4Α

-4

-2

2

4

Im Λ


PT -symmetric waveguide

−∆

∂2Ψ(x1, a) + iα(x1)Ψ(x1, a) = 0

∂2Ψ(x1,−a) + iα(x1)Ψ(x1,−a) = 0

Operator

H = −∆

Dom (H) = W 2,2(R× (−a, a)) + BC

Summary of results [BoKr08], [KrTa08]

• m-sectorial operator• sufficient conditions for real spectrum• sufficient conditions for existence or absence of eigenvalues below

essential spectrum [µ0,∞)• complex eigenvalues (numerics, lack of variational tools)

[BoKr08] 2008 Borisov, Krejcirık, Integral Equations and Operator Theory 62

[KrTa08] 2008 Krejcirık, Tater, Journal of Physics A: Mathematical and Theoretical 41


Imaginary cubic oscillatorOperator

H = −d2

dx2 + ix3

Dom (H) = ψ ∈ L2(R) : x3ψ ∈ L1loc(R), −ψ′′ + ix3ψ ∈ L2(R)

Properties

• H is J -self-adjoint and m-accretive [EdEv87]

• H has a compact resolvent (Hilbert-Schmift) [CaGrMa80], trace-class [Me00]

• spectrum is real [DoDuTa01], [Sh02]

QuestionDoes metric operator exists, i.e. is H similar to a self-adjoint operator?

[EdEv87] 1987 Edmund, Evans: Spectral Theory and Differential Operators[CaGrMa80] 1980 Caliceti, Grecchi, Maioli, Communications in Mathematical Physics 75[Me00] 2000 Mezincescu, Journal of Physics A: Mathematical and General 33[DoDuTa01] 2001 Dorey, Dunning, Tateo: Journal of Physics A: Mathematical and General 34.[Sh02] 2002 Shin, Communications in Mathematical Physics 229.


Similarity to self-adjoint operator

Theorem [SiKr12]

Eigenfunctions of H (together with associated functions) are complete in L2(R).H is not similar to any self-adjoint operator.

Proof and remarks

• completeness:• trace class resolvent and m-accretivity• abstract completeness result [GoGoKa90]

• similarity:• by contradiction: bound ‖(H − z)−1‖ ≤ K/dist(z, σ(H)) cannot

hold• semiclassical setting: Hh = −h2 d2

dx2 + ix3

• construction of quasi-modes: norm of the resolvent of Hh divergesfaster than any power of h−1 [Da99]

[GoGoKa90] 1990 Gohberg, Goldberg, Kaashoek: Classes of Linear Operators Vol. I[Da99] 1999 Davies, Communications in Mathematical Physics 200


Pseudospectrum

• σε(H) := λ ∈ C : ‖(H − λ‖ > 1/ε)• for non-self-adjoint operators pseudospectra more relevant than spectra• spectral instabilities: σε(H) =

⋃‖V‖<ε σ(H + V )

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

• work in progress with D. Krejcirık and M. Tater


Summary

Summary

• PT -symmetric operators• local non-self-adjoint Sturm-Liouville operators ⇔ non-local self-adjoint

(normal) operators• explicitly solvable models, closed formulae for Ω, Θ, h

• 2D models• strips in curved manifolds - curvature effects• waveguides - both essential and discrete spectrum

• imaginary cubic oscillator• completeness of eigenfunctions• no similarity to self-adjoint operator• non-trivial pseudospectrum


Concluding remarks

Open problems

• ESF Exploratory WorkshopMathematical Aspects of Physics with non-self-adjoint operatorsPrague, August 30 - September 4, 2010

• www.ujf.cas.cz/ESFxNSA/• list of open problems with non-self-adjoint operators• open problems published in Integral Equations and Operator Theory

Documents

Spectral analysis of some non-self-adjoint operatorsgemma.ujf.cas.cz/~siegl/Data/pdf/ConfContr/Toulouse/2012_Siegl... · Spectral analysis of some non-self-adjoint operators Petr