Fabio Bagarello D.E.I.M., Universit` a di Palermo Prague, June - - PowerPoint PPT Presentation

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Bi-coherent states for pseudo-bosons and other stuff

Fabio Bagarello

D.E.I.M., Universit` a di Palermo

Prague, June 9-th, 2016 To Milosh, for his never-ending energy

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Organization of the talk

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Organization of the talk

1

General settings: D-pseudo bosons

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Organization of the talk

1

General settings: D-pseudo bosons

2

Bicoherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Organization of the talk

1

General settings: D-pseudo bosons

2

Bicoherent states

3

Non linear extension of D-pseudo bosons

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Organization of the talk

1

General settings: D-pseudo bosons

2

Bicoherent states

3

Non linear extension of D-pseudo bosons

4

An application to the Black-Scholes equation

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Organization of the talk

1

General settings: D-pseudo bosons

2

Bicoherent states

3

Non linear extension of D-pseudo bosons

4

An application to the Black-Scholes equation

5

Conclusions

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

D-pseudo bosons

Let a and b be two operators on H, a† and b† their adjoint, and let D, dense in H, be such that a♯D ⊆ D and b♯D ⊆ D, (x♯ = x, x†). In general D ⊆ D(a♯) and D ⊆ D(b♯).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

D-pseudo bosons

Let a and b be two operators on H, a† and b† their adjoint, and let D, dense in H, be such that a♯D ⊆ D and b♯D ⊆ D, (x♯ = x, x†). In general D ⊆ D(a♯) and D ⊆ D(b♯). Definition 1: The operators (a, b) are D-pseudo bosonic (D-pb) if, for all f ∈ D, we have a b f − b a f = f. (1) ([a, b] = 1 1, for simplicity).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

D-pseudo bosons

Let a and b be two operators on H, a† and b† their adjoint, and let D, dense in H, be such that a♯D ⊆ D and b♯D ⊆ D, (x♯ = x, x†). In general D ⊆ D(a♯) and D ⊆ D(b♯). Definition 1: The operators (a, b) are D-pseudo bosonic (D-pb) if, for all f ∈ D, we have a b f − b a f = f. (1) ([a, b] = 1 1, for simplicity). Remark:– two operators (a, b) which are not D1-pb, could still be D2-pb, if a, b, D1 and D2 are chosen properly.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

D-pseudo bosons

Let a and b be two operators on H, a† and b† their adjoint, and let D, dense in H, be such that a♯D ⊆ D and b♯D ⊆ D, (x♯ = x, x†). In general D ⊆ D(a♯) and D ⊆ D(b♯). Definition 1: The operators (a, b) are D-pseudo bosonic (D-pb) if, for all f ∈ D, we have a b f − b a f = f. (1) ([a, b] = 1 1, for simplicity). Remark:– two operators (a, b) which are not D1-pb, could still be D2-pb, if a, b, D1 and D2 are chosen properly. We now assume that

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

D-pseudo bosons

Let a and b be two operators on H, a† and b† their adjoint, and let D, dense in H, be such that a♯D ⊆ D and b♯D ⊆ D, (x♯ = x, x†). In general D ⊆ D(a♯) and D ⊆ D(b♯). Definition 1: The operators (a, b) are D-pseudo bosonic (D-pb) if, for all f ∈ D, we have a b f − b a f = f. (1) ([a, b] = 1 1, for simplicity). Remark:– two operators (a, b) which are not D1-pb, could still be D2-pb, if a, b, D1 and D2 are chosen properly. We now assume that Assumption D-pb 1.– there exists a non-zero ϕ0 ∈ D such that aϕ0 = 0.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

D-pseudo bosons

Let a and b be two operators on H, a† and b† their adjoint, and let D, dense in H, be such that a♯D ⊆ D and b♯D ⊆ D, (x♯ = x, x†). In general D ⊆ D(a♯) and D ⊆ D(b♯). Definition 1: The operators (a, b) are D-pseudo bosonic (D-pb) if, for all f ∈ D, we have a b f − b a f = f. (1) ([a, b] = 1 1, for simplicity). Remark:– two operators (a, b) which are not D1-pb, could still be D2-pb, if a, b, D1 and D2 are chosen properly. We now assume that Assumption D-pb 1.– there exists a non-zero ϕ0 ∈ D such that aϕ0 = 0. Assumption D-pb 2.– there exists a non-zero Ψ0 ∈ D such that b†Ψ0 = 0.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

D-pseudo bosons

Let a and b be two operators on H, a† and b† their adjoint, and let D, dense in H, be such that a♯D ⊆ D and b♯D ⊆ D, (x♯ = x, x†). In general D ⊆ D(a♯) and D ⊆ D(b♯). Definition 1: The operators (a, b) are D-pseudo bosonic (D-pb) if, for all f ∈ D, we have a b f − b a f = f. (1) ([a, b] = 1 1, for simplicity). Remark:– two operators (a, b) which are not D1-pb, could still be D2-pb, if a, b, D1 and D2 are chosen properly. We now assume that Assumption D-pb 1.– there exists a non-zero ϕ0 ∈ D such that aϕ0 = 0. Assumption D-pb 2.– there exists a non-zero Ψ0 ∈ D such that b†Ψ0 = 0. Now, if (a, b) satisfy Definition 1, then ϕ0 ∈ D∞(b) and Ψ0 ∈ D∞(a†). Hence...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

ϕn := 1 √ n! bnϕ0, Ψn := 1 √ n! a†nΨ0, (2) n ≥ 0, can be defined and they all belong to D. We introduce, as before, FΨ = {Ψn, n ≥ 0} and Fϕ = {ϕn, n ≥ 0}. Once again, since D is stable under the action of a♯ and b♯, we deduce that both ϕn and Ψn belong to the domains of a♯, b♯ and N♯ (here N = ba).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

ϕn := 1 √ n! bnϕ0, Ψn := 1 √ n! a†nΨ0, (2) n ≥ 0, can be defined and they all belong to D. We introduce, as before, FΨ = {Ψn, n ≥ 0} and Fϕ = {ϕn, n ≥ 0}. Once again, since D is stable under the action of a♯ and b♯, we deduce that both ϕn and Ψn belong to the domains of a♯, b♯ and N♯ (here N = ba). The following lowering and raising relations hold:        b ϕn = √n + 1ϕn+1, n ≥ 0, a ϕ0 = 0, aϕn = √n ϕn−1, n ≥ 1, a†Ψn = √n + 1Ψn+1, n ≥ 0, b†Ψ0 = 0, b†Ψn = √n Ψn−1, n ≥ 1, (3) as well as the following eigenvalue equations: Nϕn = nϕn, N†Ψn = nΨn, n ≥ 0.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

ϕn := 1 √ n! bnϕ0, Ψn := 1 √ n! a†nΨ0, (2) n ≥ 0, can be defined and they all belong to D. We introduce, as before, FΨ = {Ψn, n ≥ 0} and Fϕ = {ϕn, n ≥ 0}. Once again, since D is stable under the action of a♯ and b♯, we deduce that both ϕn and Ψn belong to the domains of a♯, b♯ and N♯ (here N = ba). The following lowering and raising relations hold:        b ϕn = √n + 1ϕn+1, n ≥ 0, a ϕ0 = 0, aϕn = √n ϕn−1, n ≥ 1, a†Ψn = √n + 1Ψn+1, n ≥ 0, b†Ψ0 = 0, b†Ψn = √n Ψn−1, n ≥ 1, (3) as well as the following eigenvalue equations: Nϕn = nϕn, N†Ψn = nΨn, n ≥ 0. A consequence: if ϕ0, Ψ0 = 1, then ϕn, Ψm = δn,m, (4) for all n, m ≥ 0.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!!

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!! (2) In particular, Fϕ is a basis for H if and only if FΨ is a basis for H while ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!! (2) In particular, Fϕ is a basis for H if and only if FΨ is a basis for H while ...... if Fϕ is complete this does not imply that FΨ is complete, too.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!! (2) In particular, Fϕ is a basis for H if and only if FΨ is a basis for H while ...... if Fϕ is complete this does not imply that FΨ is complete, too. Sometimes (i.e. in concrete physical models) it is more convenient to check the following

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!! (2) In particular, Fϕ is a basis for H if and only if FΨ is a basis for H while ...... if Fϕ is complete this does not imply that FΨ is complete, too. Sometimes (i.e. in concrete physical models) it is more convenient to check the following Assumption D-pbw 3.– Fϕ and FΨ are D-quasi bases for H.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!! (2) In particular, Fϕ is a basis for H if and only if FΨ is a basis for H while ...... if Fϕ is complete this does not imply that FΨ is complete, too. Sometimes (i.e. in concrete physical models) it is more convenient to check the following Assumption D-pbw 3.– Fϕ and FΨ are D-quasi bases for H. This means that, ∀ f, g ∈ D, f, ϕn Ψn, g = f, Ψn ϕn, g = f, g. Let us now consider a self-adjoint, invertible, operator Θ, which leaves, together with Θ−1, D invariant: ΘD ⊆ D, Θ−1D ⊆ D. Then

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!! (2) In particular, Fϕ is a basis for H if and only if FΨ is a basis for H while ...... if Fϕ is complete this does not imply that FΨ is complete, too. Sometimes (i.e. in concrete physical models) it is more convenient to check the following Assumption D-pbw 3.– Fϕ and FΨ are D-quasi bases for H. This means that, ∀ f, g ∈ D, f, ϕn Ψn, g = f, Ψn ϕn, g = f, g. Let us now consider a self-adjoint, invertible, operator Θ, which leaves, together with Θ−1, D invariant: ΘD ⊆ D, Θ−1D ⊆ D. Then Definition 2: We will say that (a, b†) are Θ−conjugate if af = Θ−1b† Θ f, for all f ∈ D. (Briefly, a = Θ−1b† Θ.)

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Assumption D-pb 3.– Fϕ is a basis for H. Remarks:– (1) for non o.n. sets completeness=basis!! (2) In particular, Fϕ is a basis for H if and only if FΨ is a basis for H while ...... if Fϕ is complete this does not imply that FΨ is complete, too. Sometimes (i.e. in concrete physical models) it is more convenient to check the following Assumption D-pbw 3.– Fϕ and FΨ are D-quasi bases for H. This means that, ∀ f, g ∈ D, f, ϕn Ψn, g = f, Ψn ϕn, g = f, g. Let us now consider a self-adjoint, invertible, operator Θ, which leaves, together with Θ−1, D invariant: ΘD ⊆ D, Θ−1D ⊆ D. Then Definition 2: We will say that (a, b†) are Θ−conjugate if af = Θ−1b† Θ f, for all f ∈ D. (Briefly, a = Θ−1b† Θ.) From now on, we assume D-pb 1, D-pb 2 and D-pbw 3. Then...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

• 1. ”Extended” harmonic oscillator(s) [1-d, 2-d, or more]

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

• 1. ”Extended” harmonic oscillator(s) [1-d, 2-d, or more]
• 2. Swanson model

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

• 1. ”Extended” harmonic oscillator(s) [1-d, 2-d, or more]
• 2. Swanson model
• 3. Non commutative 2-d systems

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

• 1. ”Extended” harmonic oscillator(s) [1-d, 2-d, or more]
• 2. Swanson model
• 3. Non commutative 2-d systems
• 4. Deformed Jaynes-Cummings Model

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

• 1. ”Extended” harmonic oscillator(s) [1-d, 2-d, or more]
• 2. Swanson model
• 3. Non commutative 2-d systems
• 4. Deformed Jaynes-Cummings Model
• 5. Generalized Bogoliubov transformations

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

• 1. ”Extended” harmonic oscillator(s) [1-d, 2-d, or more]
• 2. Swanson model
• 3. Non commutative 2-d systems
• 4. Deformed Jaynes-Cummings Model
• 5. Generalized Bogoliubov transformations
• 6. Deformed graphene

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Proposition: The operators (a, b†) are Θ−conjugate if and only if Ψn = Θϕn, for all n ≥ 0. Moreover, if (a, b†) are Θ−conjugate, then (i) f, Θf > 0 for all non zero f ∈ D(Θ), and (ii) Ng = Θ−1N†Θg, for all g ∈ D. Applications:–

• 1. ”Extended” harmonic oscillator(s) [1-d, 2-d, or more]
• 2. Swanson model
• 3. Non commutative 2-d systems
• 4. Deformed Jaynes-Cummings Model
• 5. Generalized Bogoliubov transformations
• 6. Deformed graphene

.... and more

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

”Standard” coherent states (SCSs):

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

”Standard” coherent states (SCSs): Let [c, c†] = 1 1, ce0 = 0, ek =

1 √ k! c†ke0, k ≥ 0 and

W(z) = ezc†−z c, a standard coherent state is the vector Φ(z) = W(z)e0 = e−|z|2/2

∞

• k=0

zk √ k! ek. The vector Φ(z) is well defined (i.e., the series converge), and normalized ∀ z ∈ C. In fact W(z) is unitary (or ek, el = δk,l).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

”Standard” coherent states (SCSs): Let [c, c†] = 1 1, ce0 = 0, ek =

1 √ k! c†ke0, k ≥ 0 and

W(z) = ezc†−z c, a standard coherent state is the vector Φ(z) = W(z)e0 = e−|z|2/2

∞

• k=0

zk √ k! ek. The vector Φ(z) is well defined (i.e., the series converge), and normalized ∀ z ∈ C. In fact W(z) is unitary (or ek, el = δk,l). Moreover, c Φ(z) = zΦ(z), and 1 π ˆ

C

d2z|Φ(z) Φ(z)| = 1 1. It is also well known that Φ(z) saturates the Heisenberg uncertainty relation: ∆x∆p = 1

2 .

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems: for instance,

1 π

´

C |Φ(z) >< Φ(z)| z dz = c

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems: for instance,

1 π

´

C |Φ(z) >< Φ(z)| z dz = c

3

quantum information

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems: for instance,

1 π

´

C |Φ(z) >< Φ(z)| z dz = c

3

quantum information

4

several quantum potentials ⇆ CSs

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems: for instance,

1 π

´

C |Φ(z) >< Φ(z)| z dz = c

3

quantum information

4

several quantum potentials ⇆ CSs Moreover:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems: for instance,

1 π

´

C |Φ(z) >< Φ(z)| z dz = c

3

quantum information

4

several quantum potentials ⇆ CSs Moreover: several extensions of CSs do exist (Vector CSs, Gazeau-Klauder CSs,...)

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems: for instance,

1 π

´

C |Φ(z) >< Φ(z)| z dz = c

3

quantum information

4

several quantum potentials ⇆ CSs Moreover: several extensions of CSs do exist (Vector CSs, Gazeau-Klauder CSs,...) (see also squeezed states and wavelets)

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Few applications of SCSs:

1

quantum ⇒ classical they were proposed as the ”most classical” among all the possible quantum states

2

classical ⇒ quantum they are used to quantize systems: for instance,

1 π

´

C |Φ(z) >< Φ(z)| z dz = c

3

quantum information

4

several quantum potentials ⇆ CSs Moreover: several extensions of CSs do exist (Vector CSs, Gazeau-Klauder CSs,...) (see also squeezed states and wavelets) for a recent review, see the special issue in J. Phys. A, Coherent states: mathematical and physical aspects, 2012, Edited by S T. Ali, J.-P. Antoine, F. Bagarello and J.-P. Gazeau

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now: [c, c†] = 1 1 is replaced by [a, b] = 1 1.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now: [c, c†] = 1 1 is replaced by [a, b] = 1 1. Then c − → a, c† − → b, and therefore

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now: [c, c†] = 1 1 is replaced by [a, b] = 1 1. Then c − → a, c† − → b, and therefore W(z) = ezc†−z c − → U(z) = ez b−z a

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now: [c, c†] = 1 1 is replaced by [a, b] = 1 1. Then c − → a, c† − → b, and therefore W(z) = ezc†−z c − → U(z) = ez b−z a But, while W(z) is unitary (and W(z) = 1: W(z) ∈ B(H)), U(z) is not. In particular, it could easily be unbounded, at least for some z ∈ C.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now: [c, c†] = 1 1 is replaced by [a, b] = 1 1. Then c − → a, c† − → b, and therefore W(z) = ezc†−z c − → U(z) = ez b−z a But, while W(z) is unitary (and W(z) = 1: W(z) ∈ B(H)), U(z) is not. In particular, it could easily be unbounded, at least for some z ∈ C. Moreover: since b and a† are both raising operators, while b† and a are both lowering

• perators, instead of U(z) we can introduce the second operator

U(z) = ez b−z a − → V (z) = ez a†−z b†.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now: [c, c†] = 1 1 is replaced by [a, b] = 1 1. Then c − → a, c† − → b, and therefore W(z) = ezc†−z c − → U(z) = ez b−z a But, while W(z) is unitary (and W(z) = 1: W(z) ∈ B(H)), U(z) is not. In particular, it could easily be unbounded, at least for some z ∈ C. Moreover: since b and a† are both raising operators, while b† and a are both lowering

• perators, instead of U(z) we can introduce the second operator

U(z) = ez b−z a − → V (z) = ez a†−z b†. Notice that V (z) =

• U−1(z)

† .

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

But now: [c, c†] = 1 1 is replaced by [a, b] = 1 1. Then c − → a, c† − → b, and therefore W(z) = ezc†−z c − → U(z) = ez b−z a But, while W(z) is unitary (and W(z) = 1: W(z) ∈ B(H)), U(z) is not. In particular, it could easily be unbounded, at least for some z ∈ C. Moreover: since b and a† are both raising operators, while b† and a are both lowering

• perators, instead of U(z) we can introduce the second operator

U(z) = ez b−z a − → V (z) = ez a†−z b†. Notice that V (z) =

• U−1(z)

† . Notice that: these are formal equalities and definitions. In fact:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

2

Do they produce coherent states?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

2

Do they produce coherent states?

3

If this is the case, which are the properties of these states?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

2

Do they produce coherent states?

3

If this is the case, which are the properties of these states? The ones constructed using U(z) are somehow related to the ones constructed using V (z)?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

2

Do they produce coherent states?

3

If this is the case, which are the properties of these states? The ones constructed using U(z) are somehow related to the ones constructed using V (z)?

4

What does it happen if U(z) and V (z) are unbounded?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

2

Do they produce coherent states?

3

If this is the case, which are the properties of these states? The ones constructed using U(z) are somehow related to the ones constructed using V (z)?

4

What does it happen if U(z) and V (z) are unbounded?

5

Are they densely defined, in this case? Or, at least,....

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

2

Do they produce coherent states?

3

If this is the case, which are the properties of these states? The ones constructed using U(z) are somehow related to the ones constructed using V (z)?

4

What does it happen if U(z) and V (z) are unbounded?

5

Are they densely defined, in this case? Or, at least,....

6

... do they have an interesting domain?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Standard coherent states, first

1

Are U(z) and V (z) bounded in some cases?

2

Do they produce coherent states?

3

If this is the case, which are the properties of these states? The ones constructed using U(z) are somehow related to the ones constructed using V (z)?

4

What does it happen if U(z) and V (z) are unbounded?

5

Are they densely defined, in this case? Or, at least,....

6

... do they have an interesting domain?

7

and many other... (i.e. applications, concrete examples, quantization,....)

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:–

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:– is it possible to construct coherent states attached to a and b, if [a, b] = 1 1?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:– is it possible to construct coherent states attached to a and b, if [a, b] = 1 1? Answer:– yes, but it could be not entirely trivial, except if Assumption D − pb3 before is replaced by its stronger version:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:– is it possible to construct coherent states attached to a and b, if [a, b] = 1 1? Answer:– yes, but it could be not entirely trivial, except if Assumption D − pb3 before is replaced by its stronger version: Assumption D-pbs 3.– Fϕ is a Riesz basis for H.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:– is it possible to construct coherent states attached to a and b, if [a, b] = 1 1? Answer:– yes, but it could be not entirely trivial, except if Assumption D − pb3 before is replaced by its stronger version: Assumption D-pbs 3.– Fϕ is a Riesz basis for H. Then a pair (S, Fe = {en, n ≥ 0}) exists, with S, S−1 ∈ B(H), such that ϕn = Sen. FΨ is also a Riesz basis for H, and Ψn = (S−1)†en.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:– is it possible to construct coherent states attached to a and b, if [a, b] = 1 1? Answer:– yes, but it could be not entirely trivial, except if Assumption D − pb3 before is replaced by its stronger version: Assumption D-pbs 3.– Fϕ is a Riesz basis for H. Then a pair (S, Fe = {en, n ≥ 0}) exists, with S, S−1 ∈ B(H), such that ϕn = Sen. FΨ is also a Riesz basis for H, and Ψn = (S−1)†en. Now, putting Θ := (S†S)−1, then Θ, Θ−1 ∈ B(H), are self-adjoint, positive, and Ψn = Θϕn. Moreover

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:– is it possible to construct coherent states attached to a and b, if [a, b] = 1 1? Answer:– yes, but it could be not entirely trivial, except if Assumption D − pb3 before is replaced by its stronger version: Assumption D-pbs 3.– Fϕ is a Riesz basis for H. Then a pair (S, Fe = {en, n ≥ 0}) exists, with S, S−1 ∈ B(H), such that ϕn = Sen. FΨ is also a Riesz basis for H, and Ψn = (S−1)†en. Now, putting Θ := (S†S)−1, then Θ, Θ−1 ∈ B(H), are self-adjoint, positive, and Ψn = Θϕn. Moreover Θ =

∞

• n=0

|Ψn Ψn|, Θ−1 =

∞

• n=0

|ϕn ϕn|.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Question:– is it possible to construct coherent states attached to a and b, if [a, b] = 1 1? Answer:– yes, but it could be not entirely trivial, except if Assumption D − pb3 before is replaced by its stronger version: Assumption D-pbs 3.– Fϕ is a Riesz basis for H. Then a pair (S, Fe = {en, n ≥ 0}) exists, with S, S−1 ∈ B(H), such that ϕn = Sen. FΨ is also a Riesz basis for H, and Ψn = (S−1)†en. Now, putting Θ := (S†S)−1, then Θ, Θ−1 ∈ B(H), are self-adjoint, positive, and Ψn = Θϕn. Moreover Θ =

∞

• n=0

|Ψn Ψn|, Θ−1 =

∞

• n=0

|ϕn ϕn|. Of course both |Ψn Ψn| and |ϕn ϕn| are not projection operators, since Ψn, ϕn = 1, in general. They are rank-one operators.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Now, because of the Baker-Campbell-Hausdorff formula, we can write U(z) = ezb−z a = e−|z|2/2 ez b e−z a, V (z) = eza†−z b† = e−|z|2/2 ez a† e−z b†.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Now, because of the Baker-Campbell-Hausdorff formula, we can write U(z) = ezb−z a = e−|z|2/2 ez b e−z a, V (z) = eza†−z b† = e−|z|2/2 ez a† e−z b†. Of course, if a = b†, then U(z) = V (z) (= W(z)) and the operator is unitary. Let now put ϕ(z) = U(z)ϕ0, Ψ(z) = V (z) Ψ0. (5) Under Assumption D-pbs 3, these are both well defined for all z ∈ C. This is not granted, now, since U(z) and V (z) are unbounded operators (i.e., it may be that ϕ0 / ∈ D(U(z)) and/or Ψ0 / ∈ D(V (z))).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Now, because of the Baker-Campbell-Hausdorff formula, we can write U(z) = ezb−z a = e−|z|2/2 ez b e−z a, V (z) = eza†−z b† = e−|z|2/2 ez a† e−z b†. Of course, if a = b†, then U(z) = V (z) (= W(z)) and the operator is unitary. Let now put ϕ(z) = U(z)ϕ0, Ψ(z) = V (z) Ψ0. (5) Under Assumption D-pbs 3, these are both well defined for all z ∈ C. This is not granted, now, since U(z) and V (z) are unbounded operators (i.e., it may be that ϕ0 / ∈ D(U(z)) and/or Ψ0 / ∈ D(V (z))). However, this is not the case here, since ϕn = Sen ≤ S and Ψn = (S−1)†en ≤ S−1, so that ϕ(z) = e−|z|2/2

∞

• n=0

zn √ n! ϕn, Ψ(z) = e−|z|2/2

∞

• n=0

zn √ n! Ψn. converge for all z ∈ C. Hence both ϕ(z) and Ψ(z) are defined everywhere in the complex plane.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

RBCSs are related to CSs as Riesz bases are related to orthonormal bases:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

RBCSs are related to CSs as Riesz bases are related to orthonormal bases: first we deduce that:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

RBCSs are related to CSs as Riesz bases are related to orthonormal bases: first we deduce that: U(z)f = SW(z)S−1f, and V (z)f = (S−1)†W(z)S†f for all f ∈ D,

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

RBCSs are related to CSs as Riesz bases are related to orthonormal bases: first we deduce that: U(z)f = SW(z)S−1f, and V (z)f = (S−1)†W(z)S†f for all f ∈ D, and secondly we find that

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

RBCSs are related to CSs as Riesz bases are related to orthonormal bases: first we deduce that: U(z)f = SW(z)S−1f, and V (z)f = (S−1)†W(z)S†f for all f ∈ D, and secondly we find that ϕ(z) = U(z)ϕ0 = SΦ(z), Ψ(z) = V Ψ0 = (S−1)†Φ(z), for all z ∈ C.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

RBCSs are related to CSs as Riesz bases are related to orthonormal bases: first we deduce that: U(z)f = SW(z)S−1f, and V (z)f = (S−1)†W(z)S†f for all f ∈ D, and secondly we find that ϕ(z) = U(z)ϕ0 = SΦ(z), Ψ(z) = V Ψ0 = (S−1)†Φ(z), for all z ∈ C. This suggests the following generalization, which extends the definition of Riesz bases:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

RBCSs are related to CSs as Riesz bases are related to orthonormal bases: first we deduce that: U(z)f = SW(z)S−1f, and V (z)f = (S−1)†W(z)S†f for all f ∈ D, and secondly we find that ϕ(z) = U(z)ϕ0 = SΦ(z), Ψ(z) = V Ψ0 = (S−1)†Φ(z), for all z ∈ C. This suggests the following generalization, which extends the definition of Riesz bases: A pair of vectors (η(z), ξ(z)), z ∈ E, for some E ⊆ C, are called Riesz bicoherent states (RBCSs) if a standard coherent state Φ(z), z ∈ E, and a bounded operator T with bounded inverse T −1 exists such that η(z) = TΦ(z), ξ(z) = (T −1)†Φ(z),

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Then (ϕ(z), Ψ(z)) are RBCSs, with E = C.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Then (ϕ(z), Ψ(z)) are RBCSs, with E = C. RBCSs have a series of nice properties, which follow easily from similar properties of Φ(z):

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Then (ϕ(z), Ψ(z)) are RBCSs, with E = C. RBCSs have a series of nice properties, which follow easily from similar properties of Φ(z): Let (η(z), ξ(z)), z ∈ C, be a pair of RBCSs. Then:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Then (ϕ(z), Ψ(z)) are RBCSs, with E = C. RBCSs have a series of nice properties, which follow easily from similar properties of Φ(z): Let (η(z), ξ(z)), z ∈ C, be a pair of RBCSs. Then: (1) η(z), ξ(z) = 1, ∀ z ∈ C.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Then (ϕ(z), Ψ(z)) are RBCSs, with E = C. RBCSs have a series of nice properties, which follow easily from similar properties of Φ(z): Let (η(z), ξ(z)), z ∈ C, be a pair of RBCSs. Then: (1) η(z), ξ(z) = 1, ∀ z ∈ C. (2) For all f, g ∈ H the following equality (resolution of the identity) holds: f, g = 1 π ˆ

C

d2z f, η(z) ξ(z), g

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Then (ϕ(z), Ψ(z)) are RBCSs, with E = C. RBCSs have a series of nice properties, which follow easily from similar properties of Φ(z): Let (η(z), ξ(z)), z ∈ C, be a pair of RBCSs. Then: (1) η(z), ξ(z) = 1, ∀ z ∈ C. (2) For all f, g ∈ H the following equality (resolution of the identity) holds: f, g = 1 π ˆ

C

d2z f, η(z) ξ(z), g (3) If a subset D ⊂ H exists, dense in H and invariant under the action of T ♯, (T −1)♯ and c♯, and if the standard coherent state Φ(z) belongs to D, then two operators a and b exist, satisfying [a, b] = 1 1, such that a η(z) = zη(z), b†ξ(z) = zξ(z)

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

An example from the harmonic oscillator:– In this case the SCS is Φz(x) = 1 π1/4 e− 1

2 x2+

√ 2zx−ℜ(z)2.

Now, let P = |e0 e0| be the orthogonal projector operator on the ground state e0(x) =

1 π1/4 e− 1

2 x2 of the harmonic oscillator. Then the operator T = 1

1 + iP is bounded, invertible, and its inverse, T −1 = 1 1 − 1+i

2

P, is also bounded. Hence ϕz(x) = TΦz(x) = e0(x)

• e

√ 2zx−ℜ(z)2 + ie− 1

2 |z|2+ i 2 ℜ(z)ℑ(z)

=, = i Φz(x)e− 1

2 |z|2+ i 2 ℜ(z)ℑ(z)

and Ψz(x) = (T −1)†Φz(x) = e0(x)

• e

√ 2zx−ℜ(z)2 − 1 − i

2 e− 1

2 |z|2+ i 2 ℜ(z)ℑ(z)

• =

= − 1 − i 2 Φz(x)e− 1

2 |z|2+ i 2 ℜ(z)ℑ(z),

are our RBCSs, in coordinate representation.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Riesz bi-coherent states

An example from the harmonic oscillator:– In this case the SCS is Φz(x) = 1 π1/4 e− 1

2 x2+

√ 2zx−ℜ(z)2.

Now, let P = |e0 e0| be the orthogonal projector operator on the ground state e0(x) =

1 π1/4 e− 1

2 x2 of the harmonic oscillator. Then the operator T = 1

1 + iP is bounded, invertible, and its inverse, T −1 = 1 1 − 1+i

2

P, is also bounded. Hence ϕz(x) = TΦz(x) = e0(x)

• e

√ 2zx−ℜ(z)2 + ie− 1

2 |z|2+ i 2 ℜ(z)ℑ(z)

=, = i Φz(x)e− 1

2 |z|2+ i 2 ℜ(z)ℑ(z)

and Ψz(x) = (T −1)†Φz(x) = e0(x)

• e

√ 2zx−ℜ(z)2 − 1 − i

2 e− 1

2 |z|2+ i 2 ℜ(z)ℑ(z)

• =

= − 1 − i 2 Φz(x)e− 1

2 |z|2+ i 2 ℜ(z)ℑ(z),

are our RBCSs, in coordinate representation. Of course, more examples can be deduced by replacing P with some different

• rthogonal projector, for instance with the projector on a given normalized vector

u(x), Pu = |u u|, u(x) = e0(x).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Question:– in many concrete physical cases, we do not have Riesz bases but only (D-quasi) bases of biorthogonal vectors satisfying the following lowering relations: Aϕk = 0, if k = 0 √εkϕk−1 if k ≥ 1, and B†ψk = 0, if k = 0 √εkψk−1 if k ≥ 1, for some 0 = ε0 < ε1 < ε2 < · · · , or some specular raising relations. Here (Fϕ, Fψ, εn) are the main ingredients, while A and B can be ”defined” out of them.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Question:– in many concrete physical cases, we do not have Riesz bases but only (D-quasi) bases of biorthogonal vectors satisfying the following lowering relations: Aϕk = 0, if k = 0 √εkϕk−1 if k ≥ 1, and B†ψk = 0, if k = 0 √εkψk−1 if k ≥ 1, for some 0 = ε0 < ε1 < ε2 < · · · , or some specular raising relations. Here (Fϕ, Fψ, εn) are the main ingredients, while A and B can be ”defined” out of them. Then, what can we do? In particulars...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Question:– in many concrete physical cases, we do not have Riesz bases but only (D-quasi) bases of biorthogonal vectors satisfying the following lowering relations: Aϕk = 0, if k = 0 √εkϕk−1 if k ≥ 1, and B†ψk = 0, if k = 0 √εkψk−1 if k ≥ 1, for some 0 = ε0 < ε1 < ε2 < · · · , or some specular raising relations. Here (Fϕ, Fψ, εn) are the main ingredients, while A and B can be ”defined” out of them. Then, what can we do? In particulars... .... does it still make sense to define, as we do for RBCSs, vectors like ϕ(z) = N(|z|)

∞

• k=0

zk √εk! ϕk, ψ(z) = N(|z|)

∞

• k=0

zk √εk! ψk, for some normalization function N(|z|) and for generic z ∈ C? (Here ε0! = 1, εk! = ε1 · · · εk, for k ≥ 1)

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Question:– in many concrete physical cases, we do not have Riesz bases but only (D-quasi) bases of biorthogonal vectors satisfying the following lowering relations: Aϕk = 0, if k = 0 √εkϕk−1 if k ≥ 1, and B†ψk = 0, if k = 0 √εkψk−1 if k ≥ 1, for some 0 = ε0 < ε1 < ε2 < · · · , or some specular raising relations. Here (Fϕ, Fψ, εn) are the main ingredients, while A and B can be ”defined” out of them. Then, what can we do? In particulars... .... does it still make sense to define, as we do for RBCSs, vectors like ϕ(z) = N(|z|)

∞

• k=0

zk √εk! ϕk, ψ(z) = N(|z|)

∞

• k=0

zk √εk! ψk, for some normalization function N(|z|) and for generic z ∈ C? (Here ε0! = 1, εk! = ε1 · · · εk, for k ≥ 1) The answer is yes, under suitable (and rather general) conditions:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Assume that: there exist four constants rϕ, rψ > 0, and 0 ≤ αϕ, αψ ≤

1 2 , such that ϕn ≤

rn

ϕ(εn!)αϕ and ψn ≤ rn ψ(εn!)αψ, for all n ≥ 0. Let us define

ρϕ = 1 rϕ lim

k (εk+1)1/2−αϕ,

ρψ = 1 rψ lim

k (εk+1)1/2−αψ,

ˆ ρ = lim

k εk+1,

and ρ := min

• ρϕ, ρψ, √ˆ

ρ

• . Let Cρ(0) be the circle in the complex plane centered in

the origin and with radius ρ. Then....

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Assume that: there exist four constants rϕ, rψ > 0, and 0 ≤ αϕ, αψ ≤

1 2 , such that ϕn ≤

rn

ϕ(εn!)αϕ and ψn ≤ rn ψ(εn!)αψ, for all n ≥ 0. Let us define

ρϕ = 1 rϕ lim

k (εk+1)1/2−αϕ,

ρψ = 1 rψ lim

k (εk+1)1/2−αψ,

ˆ ρ = lim

k εk+1,

and ρ := min

• ρϕ, ρψ, √ˆ

ρ

• . Let Cρ(0) be the circle in the complex plane centered in

the origin and with radius ρ. Then.... ....calling N(|z|) = ∞

k=0 |z|2k εk!

−1/2 and ϕ(z) = N(|z|)

∞

• k=0

zk √εk! ϕk, ψ(z) = N(|z|)

∞

• k=0

zk √εk! ψk,

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

then.... ...these ϕ(z) and ψ(z) are well defined for z ∈ Cρ(0).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

then.... ...these ϕ(z) and ψ(z) are well defined for z ∈ Cρ(0). Moreover, for all such z, ϕ(z), ψ(z) = 1, Aϕ(z) = z ϕ(z) and B†ψ(z) = z ψ(z).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

then.... ...these ϕ(z) and ψ(z) are well defined for z ∈ Cρ(0). Moreover, for all such z, ϕ(z), ψ(z) = 1, Aϕ(z) = z ϕ(z) and B†ψ(z) = z ψ(z). Also, if a measure dλ(r) exists such that ´ ρ

0 dλ(r)r2k = εk! 2π , for all k ≥ 0, then, calling

dν(z, z) = dλ(r) dθ, we have ˆ

Cρ(0)

dν(z, z)N(|z|)−2 f, ϕ(z) ψ(z), g = f, g , for all f, g ∈ H.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;
• 2. in particular, the situation is really friendly for RBCSs

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;
• 2. in particular, the situation is really friendly for RBCSs
• 3. of course, the saturation of the Heisenberg inequality looks slightly different.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;
• 2. in particular, the situation is really friendly for RBCSs
• 3. of course, the saturation of the Heisenberg inequality looks slightly different.
• 4. they can be used as quantization tools.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;
• 2. in particular, the situation is really friendly for RBCSs
• 3. of course, the saturation of the Heisenberg inequality looks slightly different.
• 4. they can be used as quantization tools.

Open points:

• 1. if Fϕ and FΨ are only D-quasi bases, at most the BCSs produce a resolution of the

identity in D.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;
• 2. in particular, the situation is really friendly for RBCSs
• 3. of course, the saturation of the Heisenberg inequality looks slightly different.
• 4. they can be used as quantization tools.

Open points:

• 1. if Fϕ and FΨ are only D-quasi bases, at most the BCSs produce a resolution of the

identity in D.

• 2. Maybe, for physical applications, it could be enough.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;
• 2. in particular, the situation is really friendly for RBCSs
• 3. of course, the saturation of the Heisenberg inequality looks slightly different.
• 4. they can be used as quantization tools.

Open points:

• 1. if Fϕ and FΨ are only D-quasi bases, at most the BCSs produce a resolution of the

identity in D.

• 2. Maybe, for physical applications, it could be enough.
• 3. A mathematical aspect of BCSs: overcompleteness versus completeness

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

NON-Riesz bi-coherent states

Summarizing we have: Nice results

• 1. under the above assumptions, we still find most of the properties of SCSs;
• 2. in particular, the situation is really friendly for RBCSs
• 3. of course, the saturation of the Heisenberg inequality looks slightly different.
• 4. they can be used as quantization tools.

Open points:

• 1. if Fϕ and FΨ are only D-quasi bases, at most the BCSs produce a resolution of the

identity in D.

• 2. Maybe, for physical applications, it could be enough.
• 3. A mathematical aspect of BCSs: overcompleteness versus completeness
• 4. But not many examples exist, so far (but some examples do exist!).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Nonlinear D pseudo-bosons

Limitation of pseudo-bosons: the eigenvalues εn of ba are necessarily linear in n.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Nonlinear D pseudo-bosons

Limitation of pseudo-bosons: the eigenvalues εn of ba are necessarily linear in n. Let us consider a sequence {εn} such that 0 = ε0 < ε1 < · · · < εn < · · · , two operators a and b (densely) defined on H, and a set D ⊂ H which is dense in H, stable under the action of a♯ and b♯.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Nonlinear D pseudo-bosons

Limitation of pseudo-bosons: the eigenvalues εn of ba are necessarily linear in n. Let us consider a sequence {εn} such that 0 = ε0 < ε1 < · · · < εn < · · · , two operators a and b (densely) defined on H, and a set D ⊂ H which is dense in H, stable under the action of a♯ and b♯. Adopting the same general ideas used for coherent states, we will say that the triple (a, b, {εn}) is a family of non linear D pseudo-bosons (D-NLPB) if the following properties hold:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Nonlinear D pseudo-bosons

Limitation of pseudo-bosons: the eigenvalues εn of ba are necessarily linear in n. Let us consider a sequence {εn} such that 0 = ε0 < ε1 < · · · < εn < · · · , two operators a and b (densely) defined on H, and a set D ⊂ H which is dense in H, stable under the action of a♯ and b♯. Adopting the same general ideas used for coherent states, we will say that the triple (a, b, {εn}) is a family of non linear D pseudo-bosons (D-NLPB) if the following properties hold:

• p1. a non zero vector Φ0 exists in D such that a Φ0 = 0;

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Nonlinear D pseudo-bosons

Limitation of pseudo-bosons: the eigenvalues εn of ba are necessarily linear in n. Let us consider a sequence {εn} such that 0 = ε0 < ε1 < · · · < εn < · · · , two operators a and b (densely) defined on H, and a set D ⊂ H which is dense in H, stable under the action of a♯ and b♯. Adopting the same general ideas used for coherent states, we will say that the triple (a, b, {εn}) is a family of non linear D pseudo-bosons (D-NLPB) if the following properties hold:

• p1. a non zero vector Φ0 exists in D such that a Φ0 = 0;
• p2. a non zero vector η0 exists in D such that b† η0 = 0;

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Nonlinear D pseudo-bosons

Limitation of pseudo-bosons: the eigenvalues εn of ba are necessarily linear in n. Let us consider a sequence {εn} such that 0 = ε0 < ε1 < · · · < εn < · · · , two operators a and b (densely) defined on H, and a set D ⊂ H which is dense in H, stable under the action of a♯ and b♯. Adopting the same general ideas used for coherent states, we will say that the triple (a, b, {εn}) is a family of non linear D pseudo-bosons (D-NLPB) if the following properties hold:

• p1. a non zero vector Φ0 exists in D such that a Φ0 = 0;
• p2. a non zero vector η0 exists in D such that b† η0 = 0;
• p3. Calling

Φn := 1 √εn! bn Φ0, ηn := 1 √εn! a†n η0, (6) we have, for all n ≥ 0, a Φn = √εn Φn−1, b†ηn = √εn ηn−1. (7)

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Nonlinear D pseudo-bosons

Limitation of pseudo-bosons: the eigenvalues εn of ba are necessarily linear in n. Let us consider a sequence {εn} such that 0 = ε0 < ε1 < · · · < εn < · · · , two operators a and b (densely) defined on H, and a set D ⊂ H which is dense in H, stable under the action of a♯ and b♯. Adopting the same general ideas used for coherent states, we will say that the triple (a, b, {εn}) is a family of non linear D pseudo-bosons (D-NLPB) if the following properties hold:

• p1. a non zero vector Φ0 exists in D such that a Φ0 = 0;
• p2. a non zero vector η0 exists in D such that b† η0 = 0;
• p3. Calling

Φn := 1 √εn! bn Φ0, ηn := 1 √εn! a†n η0, (6) we have, for all n ≥ 0, a Φn = √εn Φn−1, b†ηn = √εn ηn−1. (7)

• p4. The set FΦ = {Φn, n ≥ 0} is a (Riesz, or D-quasi) basis for H.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Remarks:– (1) Since D is stable under the action of b and a†, it follows that Φn, ηn ∈ D, for all n ≥ 0.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Remarks:– (1) Since D is stable under the action of b and a†, it follows that Φn, ηn ∈ D, for all n ≥ 0. (2) D-PB are recovered choosing εn = n.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Remarks:– (1) Since D is stable under the action of b and a†, it follows that Φn, ηn ∈ D, for all n ≥ 0. (2) D-PB are recovered choosing εn = n. (3) If FΦ is a Riesz basis for H, the D-NLPB are called regular. It might also happen that, on the other hand, FΦ and Fη are just D-quasi bases.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Remarks:– (1) Since D is stable under the action of b and a†, it follows that Φn, ηn ∈ D, for all n ≥ 0. (2) D-PB are recovered choosing εn = n. (3) If FΦ is a Riesz basis for H, the D-NLPB are called regular. It might also happen that, on the other hand, FΦ and Fη are just D-quasi bases. (4) The set Fη = {ηn, n ≥ 0} is automatically a basis for H as well. This follows from the fact that, calling M = ba, we have MΦn = εnΦn and M†ηn = εnηn. Therefore, choosing the normalization of η0 and Φ0 in such a way η0, Φ0 = 1, Fη is biorthogonal to the basis FΦ. Then, it is possible to check that Fη is the unique basis which is biorthogonal to FΦ.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Remarks:– (1) Since D is stable under the action of b and a†, it follows that Φn, ηn ∈ D, for all n ≥ 0. (2) D-PB are recovered choosing εn = n. (3) If FΦ is a Riesz basis for H, the D-NLPB are called regular. It might also happen that, on the other hand, FΦ and Fη are just D-quasi bases. (4) The set Fη = {ηn, n ≥ 0} is automatically a basis for H as well. This follows from the fact that, calling M = ba, we have MΦn = εnΦn and M†ηn = εnηn. Therefore, choosing the normalization of η0 and Φ0 in such a way η0, Φ0 = 1, Fη is biorthogonal to the basis FΦ. Then, it is possible to check that Fη is the unique basis which is biorthogonal to FΦ. It is possible to deduce also now intertwining relations: for instance, if FΦ and Fη are related by a certain self-adjoint, invertible and, in general, unbounded operator Θ which, together with Θ−1, leaves D invariant, ηn = ΘΦn, then

• M†Θ − ΘM
• Φn = 0,

for all n.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Remarks:– (1) Since D is stable under the action of b and a†, it follows that Φn, ηn ∈ D, for all n ≥ 0. (2) D-PB are recovered choosing εn = n. (3) If FΦ is a Riesz basis for H, the D-NLPB are called regular. It might also happen that, on the other hand, FΦ and Fη are just D-quasi bases. (4) The set Fη = {ηn, n ≥ 0} is automatically a basis for H as well. This follows from the fact that, calling M = ba, we have MΦn = εnΦn and M†ηn = εnηn. Therefore, choosing the normalization of η0 and Φ0 in such a way η0, Φ0 = 1, Fη is biorthogonal to the basis FΦ. Then, it is possible to check that Fη is the unique basis which is biorthogonal to FΦ. It is possible to deduce also now intertwining relations: for instance, if FΦ and Fη are related by a certain self-adjoint, invertible and, in general, unbounded operator Θ which, together with Θ−1, leaves D invariant, ηn = ΘΦn, then

• M†Θ − ΘM
• Φn = 0,

for all n. Therefore, M satisfies an intertwining relation, and M is factorizable.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Remarks:– (1) Since D is stable under the action of b and a†, it follows that Φn, ηn ∈ D, for all n ≥ 0. (2) D-PB are recovered choosing εn = n. (3) If FΦ is a Riesz basis for H, the D-NLPB are called regular. It might also happen that, on the other hand, FΦ and Fη are just D-quasi bases. (4) The set Fη = {ηn, n ≥ 0} is automatically a basis for H as well. This follows from the fact that, calling M = ba, we have MΦn = εnΦn and M†ηn = εnηn. Therefore, choosing the normalization of η0 and Φ0 in such a way η0, Φ0 = 1, Fη is biorthogonal to the basis FΦ. Then, it is possible to check that Fη is the unique basis which is biorthogonal to FΦ. It is possible to deduce also now intertwining relations: for instance, if FΦ and Fη are related by a certain self-adjoint, invertible and, in general, unbounded operator Θ which, together with Θ−1, leaves D invariant, ηn = ΘΦn, then

• M†Θ − ΘM
• Φn = 0,

for all n. Therefore, M satisfies an intertwining relation, and M is factorizable. Finally, the operators a and b do not, in general, satisfy any simple commutation rule. Indeed, we can check that, for all n ≥ 0, [a, b]Φn = (εn+1 − εn) Φn, (8) which is different from [a, b] = 1 1, except if εn = n.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... to the Black-Scholes equation

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... to the Black-Scholes equation the Black-Scholes equation for option pricing with constant volatility σ is ∂C ∂t = − 1 2 σ2S2 ∂2C ∂S2 − rS ∂C ∂S − rC. Here C(S, t) is the price of the option, S is the stock price and r is the risk-free spot interest rate.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... to the Black-Scholes equation the Black-Scholes equation for option pricing with constant volatility σ is ∂C ∂t = − 1 2 σ2S2 ∂2C ∂S2 − rS ∂C ∂S − rC. Here C(S, t) is the price of the option, S is the stock price and r is the risk-free spot interest rate. Introducing S = ex, and the unknown function Ψ(x) as C(S(x), t) = eεtΨ(x), this equation (1) can be rewritten as: HBSΨ(x) = εΨ(x), HBS = − 1 2 σ2 d2 dx2 + σ2 2 − r d dx + r.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... to the Black-Scholes equation the Black-Scholes equation for option pricing with constant volatility σ is ∂C ∂t = − 1 2 σ2S2 ∂2C ∂S2 − rS ∂C ∂S − rC. Here C(S, t) is the price of the option, S is the stock price and r is the risk-free spot interest rate. Introducing S = ex, and the unknown function Ψ(x) as C(S(x), t) = eεtΨ(x), this equation (1) can be rewritten as: HBSΨ(x) = εΨ(x), HBS = − 1 2 σ2 d2 dx2 + σ2 2 − r d dx + r. Notice that HBS = H†

BS.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Defining the unbounded multiplication operator ρ = e−βx, β = 1 2 − r σ2 , this is invertible, with unbounded inverse, and it is possible to show that hBSf(x) :=

• ρHBSρ−1

f(x) =

• − 1

2 σ2 d2 dx2 + 1 2σ2 σ2 2 + r 2 f(x), where f(x) ∈ D := {f(x) ∈ S(R) : eγxf(x) ∈ S(R), ∀γ ∈ C}, a dense subset of L2(R). Here S(R) is the set of all the C∞ functions which decay to zero, together with their derivatives, faster than any inverse power.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Defining the unbounded multiplication operator ρ = e−βx, β = 1 2 − r σ2 , this is invertible, with unbounded inverse, and it is possible to show that hBSf(x) :=

• ρHBSρ−1

f(x) =

• − 1

2 σ2 d2 dx2 + 1 2σ2 σ2 2 + r 2 f(x), where f(x) ∈ D := {f(x) ∈ S(R) : eγxf(x) ∈ S(R), ∀γ ∈ C}, a dense subset of L2(R). Here S(R) is the set of all the C∞ functions which decay to zero, together with their derivatives, faster than any inverse power. Notice that hBS = h†

BS, and describes a free particle, of mass σ−2, subjected to a

constant potential. One also deduces that H†

BS = ρ2HBSρ−2: ρ2 intertwines

between HBS and H†

BS.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

[Baaquie, Roy, etc...] add to HBS some real potential V (x): Heff = HBS + V (x), used to represent a certain class of options. Of course, Heff is also not self-adjoint, but as before can be mapped into a (formally) self-adjoint operator heff = h†

eff:

heff := ρHeffρ−1 = hBS + V (x), and H†

eff = ρ2Heffρ−2.

These equalities could be made rigorous acting on functions of D. In [Jana, P. Roy,

• Phys. A, 2012] the authors were interested to the factorization of Heff:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

[Baaquie, Roy, etc...] add to HBS some real potential V (x): Heff = HBS + V (x), used to represent a certain class of options. Of course, Heff is also not self-adjoint, but as before can be mapped into a (formally) self-adjoint operator heff = h†

eff:

heff := ρHeffρ−1 = hBS + V (x), and H†

eff = ρ2Heffρ−2.

These equalities could be made rigorous acting on functions of D. In [Jana, P. Roy,

• Phys. A, 2012] the authors were interested to the factorization of Heff:

Heff can indeed be written as Heff = BA + δ, with δ := σ2β2

2

+ r, and where A and B are operators defined as follows A = σ √ 2 d dx + W(x) − β

• ,

B = σ √ 2

• − d

dx + W(x) + β

• ,

and where the real function W(x) is related to the potential V (x) as follows: V (x) = σ2 2

• W 2(x) − W ′(x)
• .

It turns out that B = ρ−2A†ρ2, i.e. that (B, A†) are ρ2-conjugate. Moreover B = A† if ρ = 1 1, which is true only if β = 0.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.1: V (x) = σ2

2

• x

σ2 + w

2 − 1

2

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.1: V (x) = σ2

2

• x

σ2 + w

2 − 1

2

In this case we find D-pseudo bosons: A = σ √ 2 d dx + x σ2 + w − β

• ,

B = σ √ 2

• − d

dx + x σ2 + w + β

• ,

so that [A, B] = 1 1 and

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.1: V (x) = σ2

2

• x

σ2 + w

2 − 1

2

In this case we find D-pseudo bosons: A = σ √ 2 d dx + x σ2 + w − β

• ,

B = σ √ 2

• − d

dx + x σ2 + w + β

• ,

so that [A, B] = 1 1 and Heff = − 1 2 σ2 d2 dx2 + σ2 2 x σ2 + w 2 + σ2 2 − r d dx + r − 1 2 = BA + δ.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.1: V (x) = σ2

2

• x

σ2 + w

2 − 1

2

In this case we find D-pseudo bosons: A = σ √ 2 d dx + x σ2 + w − β

• ,

B = σ √ 2

• − d

dx + x σ2 + w + β

• ,

so that [A, B] = 1 1 and Heff = − 1 2 σ2 d2 dx2 + σ2 2 x σ2 + w 2 + σ2 2 − r d dx + r − 1 2 = BA + δ. The eigenstates of Heff, ϕn(x), can be written as ϕn(x) = ρ−1 c†n √ n! Φ0(x) = 1

• σ 2n n!√π

Hn x σ + σw

• e− 1

2 ( x σ +σw)2

, n ∈ N0.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.1: V (x) = σ2

2

• x

σ2 + w

2 − 1

2

In this case we find D-pseudo bosons: A = σ √ 2 d dx + x σ2 + w − β

• ,

B = σ √ 2

• − d

dx + x σ2 + w + β

• ,

so that [A, B] = 1 1 and Heff = − 1 2 σ2 d2 dx2 + σ2 2 x σ2 + w 2 + σ2 2 − r d dx + r − 1 2 = BA + δ. The eigenstates of Heff, ϕn(x), can be written as ϕn(x) = ρ−1 c†n √ n! Φ0(x) = 1

• σ 2n n!√π

Hn x σ + σw

• e− 1

2 ( x σ +σw)2

, n ∈ N0. PBs can now be used to compute useful quantities...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... like the price kernel,

• x, e−τHx′

. Here |x is the generalized eigenstate of the position operator. In fact, we have two possible choices for the price kernel:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... like the price kernel,

• x, e−τHx′

. Here |x is the generalized eigenstate of the position operator. In fact, we have two possible choices for the price kernel: p1(x, x′; τ) =

• x, e−τHeff x′

, p2(x, x′; τ) =

• x, e−τH†

eff x′

• .

Assuming that Fϕ and FΨ produce a resolution of the identity, we have

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... like the price kernel,

• x, e−τHx′

. Here |x is the generalized eigenstate of the position operator. In fact, we have two possible choices for the price kernel: p1(x, x′; τ) =

• x, e−τHeff x′

, p2(x, x′; τ) =

• x, e−τH†

eff x′

• .

Assuming that Fϕ and FΨ produce a resolution of the identity, we have p1(x, x′; τ) =

∞

• n=0
• x, e−τHeff ϕn

Ψn, x′ = e−τδ+β(x−x′)

∞

• n=0

e−τnΦn(x)Φn(x′). Then

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... like the price kernel,

• x, e−τHx′

. Here |x is the generalized eigenstate of the position operator. In fact, we have two possible choices for the price kernel: p1(x, x′; τ) =

• x, e−τHeff x′

, p2(x, x′; τ) =

• x, e−τH†

eff x′

• .

Assuming that Fϕ and FΨ produce a resolution of the identity, we have p1(x, x′; τ) =

∞

• n=0
• x, e−τHeff ϕn

Ψn, x′ = e−τδ+β(x−x′)

∞

• n=0

e−τnΦn(x)Φn(x′). Then p1(x, x′; τ) = 1 σ√π e−τδ+β(x−x′)e

− 1

2

• ( x

σ +σw)2+

• x′

σ +σw

2

I(x, x′; τ), where

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

... like the price kernel,

• x, e−τHx′

. Here |x is the generalized eigenstate of the position operator. In fact, we have two possible choices for the price kernel: p1(x, x′; τ) =

• x, e−τHeff x′

, p2(x, x′; τ) =

• x, e−τH†

eff x′

• .

Assuming that Fϕ and FΨ produce a resolution of the identity, we have p1(x, x′; τ) =

∞

• n=0
• x, e−τHeff ϕn

Ψn, x′ = e−τδ+β(x−x′)

∞

• n=0

e−τnΦn(x)Φn(x′). Then p1(x, x′; τ) = 1 σ√π e−τδ+β(x−x′)e

− 1

2

• ( x

σ +σw)2+

• x′

σ +σw

2

I(x, x′; τ), where I(x, x′; τ) = 1 √ 1 − e−2τ exp x′ σ + σw 2 − 1 1 − e−2τ x′ σ + σw − e−τ x σ + σw 2 .

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.2: double knock out barrier

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.2: double knock out barrier V (x) = if x ∈]a, b[ ∞,

• therwise.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.2: double knock out barrier V (x) = if x ∈]a, b[ ∞,

• therwise.

As before we have heff = ρHeffρ−1 = hBS + V (x), where hBS = − 1 2 σ2 d2 dx2 + γ, γ := 1 2σ2 σ2 2 + r 2

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Choice nr.2: double knock out barrier V (x) = if x ∈]a, b[ ∞,

• therwise.

As before we have heff = ρHeffρ−1 = hBS + V (x), where hBS = − 1 2 σ2 d2 dx2 + γ, γ := 1 2σ2 σ2 2 + r 2 The eigensystem for heff is well known in the physical literature. We have        heffΦn(x) = ˜ εnΦn(x), n = 0, 1, 2, . . . , Φn(x) =

• 2

b−a sin [λn+1(x − a)] ,

λn =

nπ b−a ,

˜ εn =

σ2λ2

n+1

2

+ γ. This time ρ and ρ−1 are bounded (H1 = L2(a, b)), and Fϕ and FΨ defined as usual are biorthogonal Riesz bases.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

We introduce now ˆ B and ˆ A as follows: D( ˆ B) =

• f ∈ H1 :

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1 exists in H1

• ,

D( ˆ A) =

• f ∈ H1 :

∞

• n=1

√ρn Ψn, f1 ϕn−1 exists in H1

• ,

and ˆ Bf =

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1, ˆ Ag =

∞

• n=1

√ρn Ψn, g1 ϕn−1, for all f ∈ D( ˆ B) and g ∈ D( ˆ A). These two operators are densely defined since, calling Lϕ = l.s.{ϕn}, this is dense in H1, and Lϕ ⊆ D( ˆ B) and Lϕ ⊆ D( ˆ A).

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

We introduce now ˆ B and ˆ A as follows: D( ˆ B) =

• f ∈ H1 :

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1 exists in H1

• ,

D( ˆ A) =

• f ∈ H1 :

∞

• n=1

√ρn Ψn, f1 ϕn−1 exists in H1

• ,

and ˆ Bf =

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1, ˆ Ag =

∞

• n=1

√ρn Ψn, g1 ϕn−1, for all f ∈ D( ˆ B) and g ∈ D( ˆ A). These two operators are densely defined since, calling Lϕ = l.s.{ϕn}, this is dense in H1, and Lϕ ⊆ D( ˆ B) and Lϕ ⊆ D( ˆ A). In [FB, JMP, 2016], we have shown that:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

We introduce now ˆ B and ˆ A as follows: D( ˆ B) =

• f ∈ H1 :

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1 exists in H1

• ,

D( ˆ A) =

• f ∈ H1 :

∞

• n=1

√ρn Ψn, f1 ϕn−1 exists in H1

• ,

and ˆ Bf =

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1, ˆ Ag =

∞

• n=1

√ρn Ψn, g1 ϕn−1, for all f ∈ D( ˆ B) and g ∈ D( ˆ A). These two operators are densely defined since, calling Lϕ = l.s.{ϕn}, this is dense in H1, and Lϕ ⊆ D( ˆ B) and Lϕ ⊆ D( ˆ A). In [FB, JMP, 2016], we have shown that:

• ˆ

A and ˆ B are indeed D-NLPB,

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

We introduce now ˆ B and ˆ A as follows: D( ˆ B) =

• f ∈ H1 :

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1 exists in H1

• ,

D( ˆ A) =

• f ∈ H1 :

∞

• n=1

√ρn Ψn, f1 ϕn−1 exists in H1

• ,

and ˆ Bf =

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1, ˆ Ag =

∞

• n=1

√ρn Ψn, g1 ϕn−1, for all f ∈ D( ˆ B) and g ∈ D( ˆ A). These two operators are densely defined since, calling Lϕ = l.s.{ϕn}, this is dense in H1, and Lϕ ⊆ D( ˆ B) and Lϕ ⊆ D( ˆ A). In [FB, JMP, 2016], we have shown that:

• ˆ

A and ˆ B are indeed D-NLPB,

• that they factorize Heff,

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

An unusual application ...

We introduce now ˆ B and ˆ A as follows: D( ˆ B) =

• f ∈ H1 :

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1 exists in H1

• ,

D( ˆ A) =

• f ∈ H1 :

∞

• n=1

√ρn Ψn, f1 ϕn−1 exists in H1

• ,

and ˆ Bf =

∞

• n=0

√ρn+1 Ψn, f1 ϕn+1, ˆ Ag =

∞

• n=1

√ρn Ψn, g1 ϕn−1, for all f ∈ D( ˆ B) and g ∈ D( ˆ A). These two operators are densely defined since, calling Lϕ = l.s.{ϕn}, this is dense in H1, and Lϕ ⊆ D( ˆ B) and Lϕ ⊆ D( ˆ A). In [FB, JMP, 2016], we have shown that:

• ˆ

A and ˆ B are indeed D-NLPB,

• that they factorize Heff,
• the price kernel,........

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

And now?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

And now?

1

Dynamics: real and complex eigenvalues

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

And now?

1

Dynamics: real and complex eigenvalues

2

is it possible to better understand the C operator?

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

And now?

1

Dynamics: real and complex eigenvalues

2

is it possible to better understand the C operator?

3

PT-deformed graphene.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

And now?

1

Dynamics: real and complex eigenvalues

2

is it possible to better understand the C operator?

3

PT-deformed graphene.

4

Applications of bi-coherent states and more features

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

And now?

1

Dynamics: real and complex eigenvalues

2

is it possible to better understand the C operator?

3

PT-deformed graphene.

4

Applications of bi-coherent states and more features

5

Mathematics of the same problems, but in infinite-dimensional Hilbert spaces.

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

And now?

1

Dynamics: real and complex eigenvalues

2

is it possible to better understand the C operator?

3

PT-deformed graphene.

4

Applications of bi-coherent states and more features

5

Mathematics of the same problems, but in infinite-dimensional Hilbert spaces.

6

..........

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Main (mathematically oriented) reference:

Plan D-pseudo bosons Coherent States Non-linear D PBs What else?

Just out:

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• F. Bagarello, R. Passante, C. Trapani (Eds.)

Non-Hermitian Hamiltonians in Quantum Physics

Selected Contributions from the 15th International Conference on Non-Hermitian Hamiltonians in Quantum Physics, Palermo, Italy, 18-23 May 2015 Series: Springer Proceedings in Physics, Vol. 184 This book presents the Proceedings of the 15th International Conference on Non-Hermitian Hamiltonians in Quantum Physics, held in Palermo, Italy, from 18 to 23 May 2015. Non- Hermitian operators, and non-Hermitian Hamiltonians in particular, have recently received considerable attention from both the mathematics and physics communities. There has been a growing interest in non-Hermitian Hamiltonians in quantum physics since the discovery that PT-symmetric Hamiltonians can have a real spectrum and thus a physical

• relevance. The main subjects considered in this book include: PT-symmetry in quantum

physics, PT-optics, Spectral singularities and spectral techniques, Indefinite-metric theories, Open quantum systems, Krein space methods, and Biorthogonal systems and

• applications. The book also provides a summary of recent advances in pseudo-Hermitian

Hamiltonians and PT-symmetric Hamiltonians, as well as their applications in quantum physics and in the theory of open quantum systems.