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.
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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.
Introduction S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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
Introduction S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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
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
Introduction S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 Conclusions
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 .
Introduction S-L Operators Example 2D operators ix3 Conclusions
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.
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 Conclusions
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))
Introduction S-L Operators Example 2D operators ix3 Conclusions
Closed form of Θ, Ω, Ω−1
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
Closed form of Θ, Ω, Ω−1
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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.
Introduction S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 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 S-L Operators Example 2D operators ix3 Conclusions
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.
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 Conclusions
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
Introduction S-L Operators Example 2D operators ix3 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