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

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

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Assumption D -pb 3.– F ϕ is a basis for H . Remarks:– (1) for non o.n. sets completeness � = basis!! (2) In particular, F ϕ is a basis for H if and only if F Ψ is a basis for H while ...... if F ϕ is complete this does not imply that F Ψ is complete, too. Sometimes (i.e. in concrete physical models) it is more convenient to check the following Assumption D -pbw 3.– F ϕ and F Ψ are D -quasi bases for H . This means that, ∀ f, g ∈ D , � � f, ϕ n � � Ψ n , g � = � � f, Ψ n � � ϕ n , g � = � f, g � . Let us now consider a self-adjoint, invertible, operator Θ , which leaves, together with Θ − 1 , D invariant: Θ D ⊆ D , Θ − 1 D ⊆ D . Then Definition 2: We will say that ( a, b † ) are Θ − conjugate if af = Θ − 1 b † Θ f , for all f ∈ D . (Briefly, a = Θ − 1 b † 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 Ψ n = Θ ϕ n , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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 Ψ n = Θ ϕ n , for all n ≥ 0 . Moreover, if ( a, b † ) are Θ − conjugate, then (i) � f, Θ f � > 0 for all non zero f ∈ D (Θ) , and (ii) Ng = Θ − 1 N † Θ g , 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): k ! c † k e 0 , k ≥ 0 and Let [ c, c † ] = 1 1 1 , ce 0 = 0 , e k = √ W ( z ) = e zc † − z c , a standard coherent state is the vector ∞ z k Φ( z ) = W ( z ) e 0 = e −| z | 2 / 2 � √ e k . k ! k =0 The vector Φ( z ) is well defined (i.e., the series converge), and normalized ∀ z ∈ C . In fact W ( z ) is unitary (or � e k , e l � = δ k,l ).

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first ”Standard” coherent states (SCSs): k ! c † k e 0 , k ≥ 0 and Let [ c, c † ] = 1 1 1 , ce 0 = 0 , e k = √ W ( z ) = e zc † − z c , a standard coherent state is the vector ∞ z k Φ( z ) = W ( z ) e 0 = e −| z | 2 / 2 � √ e k . k ! k =0 The vector Φ( z ) is well defined (i.e., the series converge), and normalized ∀ z ∈ C . In fact W ( z ) is unitary (or � e k , e l � = δ k,l ). Moreover, 1 ˆ d 2 z | Φ( z ) �� Φ( z ) | = 1 c Φ( z ) = z Φ( z ) , and 1 . π C 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: quantum ⇒ classical 1

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum 2

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: 2

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: for instance, 2 1 ´ C | Φ( z ) >< Φ( z ) | z dz = c π

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: for instance, 2 1 ´ C | Φ( z ) >< Φ( z ) | z dz = c π quantum information 3

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: for instance, 2 1 ´ C | Φ( z ) >< Φ( z ) | z dz = c π quantum information 3 several quantum potentials ⇆ CSs 4

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: for instance, 2 1 ´ C | Φ( z ) >< Φ( z ) | z dz = c π quantum information 3 several quantum potentials ⇆ CSs 4 Moreover:

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Few applications of SCSs: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: for instance, 2 1 ´ C | Φ( z ) >< Φ( z ) | z dz = c π quantum information 3 several quantum potentials ⇆ CSs 4 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: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: for instance, 2 1 ´ C | Φ( z ) >< Φ( z ) | z dz = c π quantum information 3 several quantum potentials ⇆ CSs 4 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: quantum ⇒ classical they were proposed as the ”most classical” among all the 1 possible quantum states classical ⇒ quantum they are used to quantize systems: for instance, 2 1 ´ C | Φ( z ) >< Φ( z ) | z dz = c π quantum information 3 several quantum potentials ⇆ CSs 4 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 . → a , c † − Then 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 . → a , c † − Then c − → b , and therefore W ( z ) = e zc † − z c U ( z ) = e z 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 . → a , c † − Then c − → b , and therefore W ( z ) = e zc † − z c U ( z ) = e z 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 . → a , c † − Then c − → b , and therefore W ( z ) = e zc † − z c U ( z ) = e z 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 operators, instead of U ( z ) we can introduce the second operator V ( z ) = e z a † − z b † . U ( z ) = e z 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 . → a , c † − Then c − → b , and therefore W ( z ) = e zc † − z c U ( z ) = e z 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 operators, instead of U ( z ) we can introduce the second operator V ( z ) = e z a † − z b † . U ( z ) = e z b − z a − → Notice that � † . U − 1 ( z ) � V ( 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 . → a , c † − Then c − → b , and therefore W ( z ) = e zc † − z c U ( z ) = e z 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 operators, instead of U ( z ) we can introduce the second operator V ( z ) = e z a † − z b † . U ( z ) = e z b − z a − → Notice that � † . U − 1 ( z ) � V ( 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 Are U ( z ) and V ( z ) bounded in some cases? 1

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Are U ( z ) and V ( z ) bounded in some cases? 1 Do they produce coherent states? 2

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Are U ( z ) and V ( z ) bounded in some cases? 1 Do they produce coherent states? 2 If this is the case, which are the properties of these states? 3

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Are U ( z ) and V ( z ) bounded in some cases? 1 Do they produce coherent states? 2 If this is the case, which are the properties of these states? The ones constructed 3 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 Are U ( z ) and V ( z ) bounded in some cases? 1 Do they produce coherent states? 2 If this is the case, which are the properties of these states? The ones constructed 3 using U ( z ) are somehow related to the ones constructed using V ( z ) ? What does it happen if U ( z ) and V ( z ) are unbounded? 4

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Are U ( z ) and V ( z ) bounded in some cases? 1 Do they produce coherent states? 2 If this is the case, which are the properties of these states? The ones constructed 3 using U ( z ) are somehow related to the ones constructed using V ( z ) ? What does it happen if U ( z ) and V ( z ) are unbounded? 4 Are they densely defined, in this case? Or, at least,.... 5

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Are U ( z ) and V ( z ) bounded in some cases? 1 Do they produce coherent states? 2 If this is the case, which are the properties of these states? The ones constructed 3 using U ( z ) are somehow related to the ones constructed using V ( z ) ? What does it happen if U ( z ) and V ( z ) are unbounded? 4 Are they densely defined, in this case? Or, at least,.... 5 ... do they have an interesting domain ? 6

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Standard coherent states, first Are U ( z ) and V ( z ) bounded in some cases? 1 Do they produce coherent states? 2 If this is the case, which are the properties of these states? The ones constructed 3 using U ( z ) are somehow related to the ones constructed using V ( z ) ? What does it happen if U ( z ) and V ( z ) are unbounded? 4 Are they densely defined, in this case? Or, at least,.... 5 ... do they have an interesting domain ? 6 and many other... (i.e. applications, concrete examples, quantization,....) 7

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 − pb 3 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 − pb 3 before is replaced by its stronger version: Assumption D -pbs 3.– F ϕ 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 − pb 3 before is replaced by its stronger version: Assumption D -pbs 3.– F ϕ is a Riesz basis for H . Then a pair ( S, F e = { e n , n ≥ 0 } ) exists, with S, S − 1 ∈ B ( H ) , such that ϕ n = Se n . F Ψ is also a Riesz basis for H , and Ψ n = ( S − 1 ) † e n .

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 − pb 3 before is replaced by its stronger version: Assumption D -pbs 3.– F ϕ is a Riesz basis for H . Then a pair ( S, F e = { e n , n ≥ 0 } ) exists, with S, S − 1 ∈ B ( H ) , such that ϕ n = Se n . F Ψ is also a Riesz basis for H , and Ψ n = ( S − 1 ) † e n . Now, putting Θ := ( S † S ) − 1 , then Θ , Θ − 1 ∈ B ( H ) , are self-adjoint, positive, and Ψ n = Θ ϕ n . 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 − pb 3 before is replaced by its stronger version: Assumption D -pbs 3.– F ϕ is a Riesz basis for H . Then a pair ( S, F e = { e n , n ≥ 0 } ) exists, with S, S − 1 ∈ B ( H ) , such that ϕ n = Se n . F Ψ is also a Riesz basis for H , and Ψ n = ( S − 1 ) † e n . Now, putting Θ := ( S † S ) − 1 , then Θ , Θ − 1 ∈ B ( H ) , are self-adjoint, positive, and Ψ n = Θ ϕ n . Moreover ∞ ∞ Θ − 1 = � � Θ = | Ψ n �� Ψ n | , | ϕ n �� ϕ n | . n =0 n =0

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 − pb 3 before is replaced by its stronger version: Assumption D -pbs 3.– F ϕ is a Riesz basis for H . Then a pair ( S, F e = { e n , n ≥ 0 } ) exists, with S, S − 1 ∈ B ( H ) , such that ϕ n = Se n . F Ψ is also a Riesz basis for H , and Ψ n = ( S − 1 ) † e n . Now, putting Θ := ( S † S ) − 1 , then Θ , Θ − 1 ∈ B ( H ) , are self-adjoint, positive, and Ψ n = Θ ϕ n . Moreover ∞ ∞ Θ − 1 = � � Θ = | Ψ n �� Ψ n | , | ϕ n �� ϕ n | . n =0 n =0 Of course both | Ψ n �� Ψ n | and | ϕ n �� ϕ n | are not projection operators, since � Ψ n � , � ϕ n � � = 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 V ( z ) = e za † − z b † = e −| z | 2 / 2 e z a † e − z b † . U ( z ) = e zb − z a = e −| z | 2 / 2 e z b e − z a ,

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 V ( z ) = e za † − z b † = e −| z | 2 / 2 e z a † e − z b † . U ( z ) = e zb − z a = e −| z | 2 / 2 e z b e − z a , Of course, if a = b † , then U ( z ) = V ( z ) ( = W ( z ) ) and the operator is unitary. Let now put ϕ ( z ) = U ( z ) ϕ 0 , Ψ( z ) = V ( z ) Ψ 0 . (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 ϕ 0 / ∈ D ( U ( z )) and/or Ψ 0 / ∈ 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 V ( z ) = e za † − z b † = e −| z | 2 / 2 e z a † e − z b † . U ( z ) = e zb − z a = e −| z | 2 / 2 e z b e − z a , Of course, if a = b † , then U ( z ) = V ( z ) ( = W ( z ) ) and the operator is unitary. Let now put ϕ ( z ) = U ( z ) ϕ 0 , Ψ( z ) = V ( z ) Ψ 0 . (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 ϕ 0 / ∈ D ( U ( z )) and/or Ψ 0 / ∈ D ( V ( z )) ). However, this is not the case here, since � ϕ n � = � Se n � ≤ � S � and � Ψ n � = � ( S − 1 ) † e n � ≤ � S − 1 � , so that ∞ ∞ z n z n ϕ ( z ) = e −| z | 2 / 2 Ψ( z ) = e −| z | 2 / 2 � � √ ϕ n , √ Ψ n . n ! n ! n =0 n =0 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 − 1 f, V ( z ) f = ( S − 1 ) † W ( z ) S † f and 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 − 1 f, V ( z ) f = ( S − 1 ) † W ( z ) S † f and 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 − 1 f, V ( z ) f = ( S − 1 ) † W ( z ) S † f and for all f ∈ D , and secondly we find that Ψ( z ) = V Ψ 0 = ( S − 1 ) † Φ( z ) , ϕ ( z ) = U ( z ) ϕ 0 = S 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 − 1 f, V ( z ) f = ( S − 1 ) † W ( z ) S † f and for all f ∈ D , and secondly we find that Ψ( z ) = V Ψ 0 = ( S − 1 ) † Φ( z ) , ϕ ( z ) = U ( z ) ϕ 0 = S 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 − 1 f, V ( z ) f = ( S − 1 ) † W ( z ) S † f and for all f ∈ D , and secondly we find that Ψ( z ) = V Ψ 0 = ( S − 1 ) † Φ( z ) , ϕ ( z ) = U ( z ) ϕ 0 = S 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 ξ ( z ) = ( T − 1 ) † Φ( z ) , η ( z ) = T 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 ˆ d 2 z � f, η ( z ) � � ξ ( z ) , g � π 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 ˆ d 2 z � f, η ( z ) � � ξ ( z ) , g � π C (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 b † ξ ( z ) = zξ ( z ) a η ( z ) = zη ( z ) ,

Plan D -pseudo bosons Coherent States Non-linear D PBs What else? Riesz bi-coherent states