Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

On the similarity of Sturm-Liouville operatorswith non-Hermitian boundary conditions to

self-adjoint and normal operators

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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Outline

1. Introduction• Object of interest• Motivation and mathematical approach

2. General results• Symmetries• Structure of similarity transformation

3. Examples• Closed formulae for similarity transformations• Possible generalizations

4. 2D operators• Strips in curved manifolds• Waveguides

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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• J is antilinear isometric involution (J = T ) [EdEv87]

• “how far” from self-adjoint behavior we are?• similarity transformations and non-local effects

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Motivation

PT -symmetric spectral problems

• H = − d2

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

[Sh02]

• H is not similar to self-adjoint operator [SiKr12]

• 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.[SiKr] 2012 Siegl, Krejcirık, arXiv:1208.1866.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Non-self-adjoint point interactions

• general PT -symmetric point interactions in L2(R) [AlFeKu02]

• parametrization of both connected and separated PT -symmetric b.c.• spectral properties of one point interaction

• particular PT -symmetric point interaction in L2(R) [AlKu05]

• spectral properties and similarity to s-a operators• resolvent criterion approach [Na84]

• examples: real spectrum, but not similar to s-a operator

• PT -symmetric interaction in L2(−a, a) [KrBiZn06]

• Robin type boundary conditions• real spectrum and similarity to s-a operator• explicitly solvable example

[AlFeKu02] 2002 Albeverio, Fei, Kurasov, Letters in Mathematical Physics 59,[AlKu05] 2005 Albeverio, Kuzhel, Journal of Physics A: Mathematical and General 38,[Na84] 1984 Naboko, Functional Analysis and its Applications 18,[KrBiZn06] 2006 Krejcirık, Bıla, Znojil, Journal of Physics A: Mathematical and General 39.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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±

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

• eigenfunctions (together with associated functions) form Riesz basis [Mi62]

• H is a discrete spectral operator [DSIII]

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

[Mi62] 1962 Mikhajlov, Doklady Akademii Nauk SSSR 114

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Structure of Ω and ΘProposition

Let all eigenvalues of H be simple and let en be an orthonormal basis ofL2(−a, a). 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 .

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Structure of Θ and Ω

Remarks

• proof: 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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

PT -symmetric modelSimplest 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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

PT -symmetric modelEigenfunctions

• 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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Θ and Ω operators

Closed form of Θ, Ω, Ω−1

• 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))

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Θ and Ω operatorsProposition (Θ)

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

• different explicit Θ’s, construction of a dense set 2011 J. Zelezny, MT

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Θ and Ω operatorsProposition (Θ)

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

• different explicit Θ’s, construction of a dense set 2011 J. Zelezny, MT

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Similar s-a operator

• h := ΩHΩ−1

• h = −∆N + α2〈χN0 , ·〉χ

N0

• rank one perturbation of −∆N

0 1 2 3 4 5Α

5

10

15

20

Λ

Remarks

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

PT -symmetric model 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

• All EVs are real and simple if β > 0. There is either one or no complexconjugated pairs of EVs if β < 0. [KrSi10]

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

PT -symmetric model II

Metric Operator

Θ = I + Kwith

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

Remarks

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

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Irregular boundary conditions

Definition of operator

• 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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Irregular boundary conditions

Definition of operator

• 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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Point interaction• 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−).

Properties of H

• real eigenvalues (nπ/2a)2n∈N• Θ and Ω known, K ,L not compact [Si08, AlKu05, KuTr11]

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

• τ = ±π/2 irregular cases, σ(H) = σp(H) = C• ∃ extension with empty resolvent set ⇔ ∃ additional fundamental

symmetry R [KuTr11]

• algebraic construction of Θ with help of I ,P,R• h := ΩHΩ−1 = −∆D

[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.

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

2D strips in curved manifolds

General geometry and interaction α, β

Under regularity conditions on α, β and f (determines the geometry):• H(α, β) is m-sectorial• H has a compact resolvent• H∗(α, β) = H(−α, β)

Additional assumptions on symmetry of manifold:• H is PT -symmetric, P and T -self-adjoint• λ ∈ σ(H)⇔ λ ∈ σ(H)

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

Summary

Summary

• general structure of Θ, Ω• local non-self-adjoint Sturm-Liouville operators ⇔ non-local self-adjoint

(normal) operators• explicitly solvable models, closed formulae for Ω, Θ, h• more general method for construction of Θ• point interaction example Θ = I + K , K not compact and h local

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

Introduction Operator Structure of Ω Model Generalizations 2D operators Conclusions

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