Return to Question

Given the following pair of operators $a$ and $a^{\dagger}$ that satisfy the usual bosonic CCR:
$$[a,a]=[a^{\dagger},a^{\dagger}] = 0;\ [a,a^{\dagger}] = 1$$
For what values of $\alpha \in\mathbb C$ are the following expressions well defined?
$$a^{\alpha} \ \text{and}\ (a^{\dagger})^{\alpha}$$
For integer $\alpha$, I know that these expressions are well defined, but I am interested in knowing what kind of constraints are imposed on $\alpha$, if such an operator must be appropriately definable. If the above expressions can be defined in precise manner, then please also provide the appropriate construction of these operators.

Some meandering background:
The actual problem I'm trying to solve has nothing to do with quantum mechanics! It is instead a reformulation of the dynamical equations of a polymer (hydrodynamically interacting with a solvent) using the rouse modes as "bosons" [JCP(45), 1966, M. Fixman, Polymer Dynamics:Boson Representation and Excluded Volume forces]. So there are no nice hamiltonians and unitary operators to work with. The dynamical equation is of the following form:
$$\dfrac{\partial{\left|\rho\right\rangle}}{\partial t}+\mathcal{L}\left|\rho\right\rangle=0$$
Now the operator $\mathcal{L}$ has a representation in the boson operators as
$$\mathcal{L}=g(a^{\dagger}a+\zeta a^{\dagger}a^{\dagger})$$
where $g$ and $\kappa$ are real constants. The kets live in a standard bosonic Fock space, and a complete basis of number states is present constructed in a fashion equivalent to the case of the harmonic oscillator. On trying to solve the eigenvalue equation for $\mathcal{L}$, one can write down the eigenstates, indexed by the eigenvalue as
$$\left|\psi;\lambda\right\rangle=exp\left(-\frac{\zeta}{2}a^{\dagger}a^{\dagger}\right)(a^{\dagger})^{\lambda}\left|0\right\rangle$$
such that $\mathcal{L}\left|\psi;\lambda\right\rangle=\lambda\left|\psi;\lambda\right\rangle$. This is where my original question comes into the forefront - Deciding on how these operators behave on being raised to arbitrary powers, I can fix the possible eigenvalues of $\mathcal{L}$. If it is possible to arrive at a conclusion on the eigenvalues of the given $\mathcal{L}$ operator in any other fashion, without resorting to this approach, then please go ahead and tell me.

Given the following pair of operators $a$ and $a^{\dagger}$ that satisfy the usual bosonic CCR:
$$[a,a]=[a^{\dagger},a^{\dagger}] = 0;\ [a,a^{\dagger}] = 1$$
For what values of $\alpha \in\mathbb C$ are the following expressions well defined?
$$a^{\alpha} \ \text{and}\ (a^{\dagger})^{\alpha}$$
For integer $\alpha$, I know that these expressions are well defined, but I am interested in knowing what kind of constraints are imposed on $\alpha$, if such an operator must be appropriately definable. If the above expressions can be defined in precise manner, then please also provide the appropriate construction of these operators.

Some meandering background:
The actual problem I'm trying to solve has nothing to do with quantum mechanics! It is instead a reformulation of the dynamical equations of a polymer (hydrodynamically interacting with a solvent) using the rouse modes as "bosons" [JCP(45), 1966, M. Fixman, Polymer Dynamics:Boson Representation and Excluded Volume forces]. So there are no nice hamiltonians and unitary operators to work with. The dynamical equation is of the following form:
$$\dfrac{\partial{\left|\rho\right\rangle}}{\partial t}+\mathcal{L}\left|\rho\right\rangle=0$$
Now the operator $\mathcal{L}$ has a representation in the boson operators as
$$\mathcal{L}=g(a^{\dagger}a+\zeta a^{\dagger}a^{\dagger})$$
where $g$ and $\zeta$ are real constants. The kets live in a standard bosonic Fock space, and a complete basis of number states is present constructed in a fashion equivalent to the case of the harmonic oscillator. On trying to solve the eigenvalue equation for $\mathcal{L}$, one can write down the eigenstates, indexed by the eigenvalue as
$$\left|\psi;\lambda\right\rangle=exp\left(-\frac{\zeta}{2}a^{\dagger}a^{\dagger}\right)(a^{\dagger})^{\lambda}\left|0\right\rangle$$
such that $\mathcal{L}\left|\psi;\lambda\right\rangle=\lambda\left|\psi;\lambda\right\rangle$. This is where my original question comes into the forefront - Deciding on how these operators behave on being raised to arbitrary powers, I can fix the possible eigenvalues of $\mathcal{L}$. If it is possible to arrive at a conclusion on the eigenvalues of the given $\mathcal{L}$ operator in any other fashion, without resorting to this approach, then please go ahead and tell me.

Given the following pair of operators $a$ and $a^{\dagger}$ that satisfy the usual bosonic CCR:
$$[a,a]=[a^{\dagger},a^{\dagger}] = 0;\ [a,a^{\dagger}] = 1$$
For what values of $\alpha \in\mathbb C$ are the following expressions well defined?
$$a^{\alpha} \ \text{and}\ (a^{\dagger})^{\alpha}$$
For integer $\alpha$, I know that these expressions are well defined, but I am interested in knowing what kind of constraints are imposed on $\alpha$, if such an operator must be appropriately definable. If the above expressions can be defined in precise manner, then please also provide the appropriate construction of these operators.

Some meandering background:
The actual problem I'm trying to solve has nothing to do with quantum mechanics! It is instead a reformulation of the dynamical equations of a polymer (hydrodynamically interacting with a solvent) using the rouse modes as "bosons" [Ref]. So there are no nice hamiltonians and unitary operators to work with. The dynamical equation is of the following form:
$$\dfrac{\partial{\left|\rho\right\rangle}}{\partial t}+\mathcal{L}\left|\rho\right\rangle=0$$
Now the operator $\mathcal{L}$ has a representation in the boson operators as
$$\mathcal{L}=g(a^{\dagger}a+\zeta a^{\dagger}a^{\dagger})$$
where $g$ and $\kappa$ are real constants. The kets live in a standard bosonic Fock space, and a complete basis of nuber states is present constructed in a fashion equivalent to the case of the harmonic oscillator. On trying to solve the eigenvalue equation for $\mathcal{L}$, one can write down the eigenstates, indexed by the eigenvalue as
$$\left|\psi;\lambda\right\rangle=exp\left(-\frac{\zeta}{2}a^{\dagger}a^{\dagger}\right)(a^{\dagger})^{\lambda}\left|0\right\rangle$$
such that $\mathcal{L}\left|\psi;\lambda\right\rangle=\lambda\left|\psi;\lambda\right\rangle$. This is where my original question comes into the forefront - Deciding on how these operators behave on being raised to arbitrary powers, I can fix the possible eigenvalues of $\mathcal{L}$. If it is possible to arrive at a conclusion on the eigenvalues of the given $\mathcal{L}$ operator in any other fashion, without resorting to this approach, then please go ahead and tell me.

Given the following pair of operators $a$ and $a^{\dagger}$ that satisfy the usual bosonic CCR:
$$[a,a]=[a^{\dagger},a^{\dagger}] = 0;\ [a,a^{\dagger}] = 1$$
For what values of $\alpha \in\mathbb C$ are the following expressions well defined?
$$a^{\alpha} \ \text{and}\ (a^{\dagger})^{\alpha}$$
For integer $\alpha$, I know that these expressions are well defined, but I am interested in knowing what kind of constraints are imposed on $\alpha$, if such an operator must be appropriately definable. If the above expressions can be defined in precise manner, then please also provide the appropriate construction of these operators.

Some meandering background:
The actual problem I'm trying to solve has nothing to do with quantum mechanics! It is instead a reformulation of the dynamical equations of a polymer (hydrodynamically interacting with a solvent) using the rouse modes as "bosons" [JCP(45), 1966, M. Fixman, Polymer Dynamics:Boson Representation and Excluded Volume forces]. So there are no nice hamiltonians and unitary operators to work with. The dynamical equation is of the following form:
$$\dfrac{\partial{\left|\rho\right\rangle}}{\partial t}+\mathcal{L}\left|\rho\right\rangle=0$$
Now the operator $\mathcal{L}$ has a representation in the boson operators as
$$\mathcal{L}=g(a^{\dagger}a+\zeta a^{\dagger}a^{\dagger})$$
where $g$ and $\kappa$ are real constants. The kets live in a standard bosonic Fock space, and a complete basis of number states is present constructed in a fashion equivalent to the case of the harmonic oscillator. On trying to solve the eigenvalue equation for $\mathcal{L}$, one can write down the eigenstates, indexed by the eigenvalue as
$$\left|\psi;\lambda\right\rangle=exp\left(-\frac{\zeta}{2}a^{\dagger}a^{\dagger}\right)(a^{\dagger})^{\lambda}\left|0\right\rangle$$
such that $\mathcal{L}\left|\psi;\lambda\right\rangle=\lambda\left|\psi;\lambda\right\rangle$. This is where my original question comes into the forefront - Deciding on how these operators behave on being raised to arbitrary powers, I can fix the possible eigenvalues of $\mathcal{L}$. If it is possible to arrive at a conclusion on the eigenvalues of the given $\mathcal{L}$ operator in any other fashion, without resorting to this approach, then please go ahead and tell me.

Given the following pair of operators $a$ and $a^{\dagger}$ that satisfy the usual bosonic CCR:
$$[a,a]=[a^{\dagger},a^{\dagger}] = 0;\ [a,a^{\dagger}] = 1$$
For what values of $\alpha \in\mathbb C$ are the following expressions well defined?
$$a^{\alpha} \ \text{and}\ (a^{\dagger})^{\alpha}$$
For integer $\alpha$, I know that these expressions are well defined, but I am interested in knowing what kind of constraints are imposed on $\alpha$, if such an operator must be appropriately definable. If the above expressions can be defined in precise manner, then please also provide the appropriate construction of these operators.

Given the following pair of operators $a$ and $a^{\dagger}$ that satisfy the usual bosonic CCR:
$$[a,a]=[a^{\dagger},a^{\dagger}] = 0;\ [a,a^{\dagger}] = 1$$
For what values of $\alpha \in\mathbb C$ are the following expressions well defined?
$$a^{\alpha} \ \text{and}\ (a^{\dagger})^{\alpha}$$
For integer $\alpha$, I know that these expressions are well defined, but I am interested in knowing what kind of constraints are imposed on $\alpha$, if such an operator must be appropriately definable. If the above expressions can be defined in precise manner, then please also provide the appropriate construction of these operators.

Some meandering background:
The actual problem I'm trying to solve has nothing to do with quantum mechanics! It is instead a reformulation of the dynamical equations of a polymer (hydrodynamically interacting with a solvent) using the rouse modes as "bosons" [Ref]. So there are no nice hamiltonians and unitary operators to work with. The dynamical equation is of the following form:
$$\dfrac{\partial{\left|\rho\right\rangle}}{\partial t}+\mathcal{L}\left|\rho\right\rangle=0$$
Now the operator $\mathcal{L}$ has a representation in the boson operators as
$$\mathcal{L}=g(a^{\dagger}a+\zeta a^{\dagger}a^{\dagger})$$
where $g$ and $\kappa$ are real constants. The kets live in a standard bosonic Fock space, and a complete basis of nuber states is present constructed in a fashion equivalent to the case of the harmonic oscillator. On trying to solve the eigenvalue equation for $\mathcal{L}$, one can write down the eigenstates, indexed by the eigenvalue as
$$\left|\psi;\lambda\right\rangle=exp\left(-\frac{\zeta}{2}a^{\dagger}a^{\dagger}\right)(a^{\dagger})^{\lambda}\left|0\right\rangle$$
such that $\mathcal{L}\left|\psi;\lambda\right\rangle=\lambda\left|\psi;\lambda\right\rangle$. This is where my original question comes into the forefront - Deciding on how these operators behave on being raised to arbitrary powers, I can fix the possible eigenvalues of $\mathcal{L}$. If it is possible to arrive at a conclusion on the eigenvalues of the given $\mathcal{L}$ operator in any other fashion, without resorting to this approach, then please go ahead and tell me.