# Mathematical Models for Unstable Quantum Systems and Gamow States

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## Abstract

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## 1. Introduction, Motivation and General Considerations

- Resonances appear in scattering processes as high bumps on the cross section. These bumps are characterized by two parameters: the resonance energy ${E}_{R}$ and the width of the bump, $\Gamma $, for which its inverse is proportional to the mean life. Generally speaking, ${E}_{R}$ is the difference in energy between the decaying system and the products of the decay [5].
- The presence of resonances is also detected by large time delays. This is the difference in the time that one particle stays in the interaction region with or without the interaction [6].
- Sudden change of the phase shift ${\delta}_{\ell}(E)$ around the resonance energy ${E}_{R}$.
- Let us consider the wave function of the decaying state $\psi (E)$ in the energy representation. If its amplitude $|\psi (E)|$ has a Lorenztian shape,$${|\psi (E)|}^{2}\approx N\frac{\Gamma}{{(E-{E}_{R})}^{2}+{\Gamma}^{2}/4},$$

- Mathematical definitions.
- The scattering matrix S (also called scattering operator or S-operator) is a very useful tool in quantum scattering theory. This is an operator that acts on the freely evolving input state and gives as a result the freely evolving output state, ${\psi}^{\mathrm{out}}=S{\psi}^{\mathrm{in}}$. The S-operator encodes the action of the interaction given by the potential V. It is customary to give the relation ${\psi}^{\mathrm{out}}=S{\psi}^{\mathrm{in}}$ either on the momentum or in the energy representation. In the first case, S is a function of the momentum, $S(k)$, in the second, of the energy, $S(E)$.Under some causality conditions [1,2], the function $S(k)$ can be analytically continued to a meromorphic function on the whole complex plane. Resonances appear as pairs of poles on the lower half of the complex plane, with the same imaginary part and the real parts equal in modulus, and with opposite signs.In the energy representation and under the same causality conditions, the S-operator is represented by a function $S(E)$, meromorphic on a two-sheeted Riemann surface. Here, resonances are pairs of complex conjugate poles on the second sheet, located at the points ${E}_{R}\pm i\Gamma /2$, where ${E}_{R}>0$ and $\Gamma $ have the same meaning as above, i.e., resonance energy and width.In principle, these resonance poles may have any multiplicity. However, although models with double pole resonances has been constructed [12], in most cases the multiplicity is equal to one.This definition usually matches with the physical definitions given above, although this is not always the case [13]. Models with resonance poles on the analytic continuation of the S-operator, not being escorted by a bump in the cross section with equal resonance energy and width have been constructed. Analogously, models having a bump in the cross section without the corresponding pole on the S-operator also exist (see [5]). Also, S-operators admitting an analytic continuation with weird properties have also been considered [14,15,16].Generally speaking, physicists prefer this definition to the next.
- As we said before, in order to have resonances, we need a Hamiltonian pair $\{{H}_{0},H={H}_{0}+V\}$, where the V is the potential responsible of the creation of a resonance. Since physical Hamiltonians are semibounded in non-relativistic quantum mechanics, we may assume that the continuous spectrum of both ${H}_{0}$ and H coincides with the positive semiaxis ${\mathbb{R}}^{+}\equiv [0,\infty )$. For simplicity, we may assume that this spectrum is simple (This is a technical detail that we are not going to explain here. See for instance [17]). We may also assume an absence of point and continuous singular spectrum (If not, just take the absolutely continuous part of the spectrum [17]).Now, consider that in the Hilbert space $\mathcal{H}$ on which both Hamiltonian act there exists a dense subspace $\mathcal{D}\subset \mathcal{H}$, such that the following functions, defined for any $\psi \in \mathcal{D}$,$${R}_{0\psi}(\lambda ):=\langle \psi |{({H}_{0}-\lambda I)}^{-1}\psi \rangle \phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{R}_{\psi}(\lambda );=\langle \psi |{(H-\lambda I)}^{-1}\psi \rangle \phantom{\rule{0.166667em}{0ex}},$$We have to underline that both functions ${R}_{0\psi}(\lambda )$ and ${R}_{\psi}(\lambda )$ are complex analytic with a branch cut on the positive semiaxis ${\mathbb{R}}^{+}$, since the inverses ${({H}_{0}-\lambda I)}^{-1}$ and ${(H-\lambda I)}^{-1}$ are not defined for $\lambda $ in the spectrum of ${H}_{0}$ and H. Nevertheless, this analytic continuation is possible from the first to the second Riemann sheet on where the function $S(E)$ is defined [1,18]. Resonance poles appear as complex conjugate on the second sheet, exactly as happens with poles of $S(E)$.Up to our knowledge, there is not a thoroughly study of models for which resonances as poles of $S(E)$ and resonance poles given by the resolvent functions coincide. This is true for some studied cases. As proven in [19], this happens for the case of the Friedrichs model, which will be mentioned later as a solvable model showing resonances.

## 2. Rigged Hilbert Spaces and Gamow Vectors

- For almost all (with respect to the measure $d\mu $) $\lambda \in \sigma (H)$, there exists a ${F}_{\lambda}\in {\mathrm{\Phi}}^{\times}$ such that $H{F}_{\lambda}=\lambda {F}_{\lambda}$. It is customary the write ${F}_{\lambda}\equiv |\lambda \rangle $.
- The operator H admits a spectral decomposition of the form$$\langle \varphi |H\psi \rangle ={\int}_{\sigma (H)}\lambda \phantom{\rule{0.166667em}{0ex}}\langle \varphi |\lambda \rangle \langle \lambda |\psi \rangle \phantom{\rule{0.166667em}{0ex}}d\mu (\lambda )\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\varphi ,\psi \in \mathrm{\Phi}\phantom{\rule{0.166667em}{0ex}},d\mu (\lambda )\phantom{\rule{0.166667em}{0ex}},$$

- Almost all $\lambda \in \sigma (H)$ verify a Eigenvalue relation of the kind $A|\lambda \rangle =\lambda \phantom{\rule{0.166667em}{0ex}}|\lambda \rangle $, which extends the Eigenvalue equation valid for the point spectrum to all spectrum. This is particularly important when H has an absolutely continuous spectrum [17]. The only difference is that, while the Eigenvectors with Eigenvalue in the continuous spectrum are vectors in $\mathcal{H}$, the Eigenvectors $|\lambda \rangle \in {\mathrm{\Phi}}^{\times}$ and $|\lambda \rangle \notin \mathcal{H}$.
- If we omit the arbitrary $\varphi ,\psi \in \mathrm{\Phi}$, we may write, for all $n\in {\mathbb{N}}_{0}$,$${H}^{n}={\int}_{\sigma (H)}{\lambda}^{n}\phantom{\rule{0.166667em}{0ex}}|\lambda \rangle \langle \lambda |\phantom{\rule{0.166667em}{0ex}}d\mu (\lambda )\phantom{\rule{0.166667em}{0ex}}.$$For $n=0$, we obtain a spectral representation of the canonical injection $I:\mathrm{\Phi}\u27fc{\mathrm{\Phi}}^{\times}$,$$I={\int}_{\sigma (H)}|\lambda \rangle \langle \lambda |\phantom{\rule{0.166667em}{0ex}}d\mu (\lambda )\phantom{\rule{0.166667em}{0ex}}.$$This canonical injection is continuous with respect to the topologies on $\mathrm{\Phi}$ and ${\mathrm{\Phi}}^{\times}$.
- There exists a unitary mapping $U:\mathcal{H}\u27fc{L}^{2}(\sigma (H),d\mu )$, such that $UH{U}^{-1}$ is the multiplication operator on ${L}^{2}(\sigma (H),d\mu )$. For each $\psi \in \mathrm{\Phi}$, $U\psi =\langle \lambda |\psi \rangle =\psi (\lambda )$. We have a new RHS of the form $U\mathrm{\Phi}\subset {L}^{2}(\sigma (H),d\mu )\subset {(U\mathrm{\Phi})}^{\times}$. This new RHS is a concrete realization of (10), where the elements of the Hilbert space are functions and the elements of the antidual ${(U\mathrm{\Phi})}^{\times}$ are generalized functions in the sense of Gelfand [30].

#### 2.1. Gamow Vectors

- The Gamow vectors belong to the duals of respective RHS, ${\mathrm{\Phi}}_{\pm}\subset \mathcal{H}\subset {\mathrm{\Phi}}_{\pm}^{\times}$, with ${\psi}_{D}\in {\mathrm{\Phi}}_{+}^{\times}$ and ${\psi}_{G}\in {\mathrm{\Phi}}_{-}^{\times}$ [53] (The convention of signs is the opposite in [52]. Nevertheless, we have considered convenient to use the present convention.). The total Hamiltonian H is extended by duality to the antiduals ${\mathrm{\Phi}}_{\pm}^{\times}$ (11). Then, the compatibility between the self adjointness of H and the presence of Eigenvectors with complex Eigenvalues is explained by the fact that these Eigenvectors do not belong to the Hilbert space $\mathcal{H}$.
- In the Introduction, we have mentioned the existence of a background, there represented by the vector state ${\psi}_{B}$, and responsible for the deviations of the exponential law. Recall that $\psi ={\psi}_{G}+{\psi}_{B}$, then, since $\psi \in \mathcal{H}\subset {\mathrm{\Phi}}_{+}$, we infer that ${\psi}_{B}\in {\mathrm{\Phi}}_{+}$. It is rather obvious that another background vector must exist in ${\mathrm{\Phi}}_{-}^{\times}$, due to the symmetry properties both RHS, ${\mathrm{\Phi}}_{\pm}\subset \mathcal{H}\subset {\mathrm{\Phi}}_{\pm}^{\times}$.
- Using the simplification according to which H has a simple absolutely continuous spectrum ${\mathbb{R}}^{+}\equiv [0,\infty )$, we may represent the RHS ${\mathrm{\Phi}}_{\pm}\subset \mathcal{H}\subset {\mathrm{\Phi}}_{\pm}^{\times}$ using a triplet where $\mathcal{H}$ is represented by ${L}^{2}({\mathbb{R}}^{+})$, and ${\mathrm{\Phi}}_{\pm}$ are represented by respective spaces of analytic functions at least on a half plane. The most used representation for ${\mathrm{\Phi}}_{\pm}$ is given by the following spaces$$\mathcal{S}\cap {\mathcal{H}}_{\mp}^{2}{|}_{{\mathbb{R}}^{+}}\phantom{\rule{0.166667em}{0ex}},$$
- (i)
- $\mathcal{S}$ is the Schwartz space of all indefinitely differentiable functions that converge to zero faster than the inverse if any polynomial.
- (ii)
- ${\mathcal{H}}_{-}^{2}$ is the space of Hardy functions on the open lower half plane. These functions, ${f}_{-}(z)$, are analytic in the lower half plane with the property that$$\underset{y>0}{sup}{\int}_{-\infty}^{\infty}{|{f}_{-}(x-iy)|}^{2}\phantom{\rule{0.166667em}{0ex}}dx<\infty \phantom{\rule{0.166667em}{0ex}}.$$The functions of their boundary values on the whole real line $\mathbb{R}$, ${f}_{-}(x)$, are square integrable and uniquely determine the function ${f}_{-}(z)$ as defined on the whole open half plane and vice-versa, ${f}_{-}(z)$ determines uniquely the boundary function ${f}_{-}(x)$ [54,55,56]. Also, $f(z)$ can be determined by its boundary values on the positive semiaxis ${\mathbb{R}}^{+}$ [57].
- (iii)
- In (16), the functions are restricted to the positive semiaxis $[0,\infty )$, so that the functions are considered as complex functions of positive real variable.
- (iv)

It is possible to construct a representation without this restriction to the positive semiaxis. In this representation, the Gamow vectors are normalizable (although outside the domain of H), so that they must be considered as members of the antidual spaces and have the Breit-Wigner energy distribution (1). However, this construction is not unitarily equivalent to ${\mathrm{\Phi}}_{\pm}\subset \mathcal{H}\subset {\mathrm{\Phi}}_{\pm}^{\times}$ [52]. The Hardy functions on the upper half plane are defined analogously and have the same properties. - The procedure for the construction of ${\mathrm{\Phi}}_{\pm}\subset \mathcal{H}\subset {\mathrm{\Phi}}_{\pm}^{\times}$ goes as follows: The spectral theorem [17] gives a unitary operator $U:\mathcal{H}\u27fc{L}^{2}({\mathbb{R}}^{+})$, such that $UH{U}^{-1}$ is the multiplication operator on ${L}^{2}({\mathbb{R}}^{+})$. Then, construct ${\mathrm{\Phi}}_{\pm}$ as$${\mathrm{\Phi}}_{\pm}:={U}^{-1}\phantom{\rule{0.166667em}{0ex}}\left[\mathcal{S}\cap {\mathcal{H}}_{\mp}^{2}{|}_{{\mathbb{R}}^{+}}\right]\phantom{\rule{0.166667em}{0ex}}.$$Once we have the topology on (16), we transport this topology to ${\mathrm{\Phi}}_{\pm}$ by the unitary mapping ${U}^{-1}$.H is continuous on both ${\mathrm{\Phi}}_{\pm}$, and $H{\mathrm{\Phi}}_{\pm}\subset {\mathrm{\Phi}}_{\pm}$. Then, we may extend H by continuity to ${\mathrm{\Phi}}_{\pm}^{\times}$ by duality using (11). Gamow vectors are defined as taking full advantage of the analyticity properties of Hardy functions and we show Eigenvectors of this extension of H to the antiduals with the Eigenvalues given in (8) and (15).
- The interest of using spaces of Hardy functions in (16) is the following: Equation (9) is only valid for positive values of time, while the second relation in (15) is only valid for negative values of time. This result has several important consequences. The time evolution of the decaying Gamow vector, ${\psi}_{D}$, is defined for positive values of time, which is compatible with the idea that this vector represents the exact exponentially decaying part of a resonance. Otherwise, the time evolution of the growing Gamow vector, ${\psi}_{G}$, increases from $-\infty <t<0$, which means that it decays to the past. This growing Gamow vector represents the same resonance as the decaying Gamow vector and it represent the same phenomenon. One is the time reversal of the other [58]. This construction is just the point of departure of another interesting formalism, the time asymmetric quantum mechanics that we do not intend to explain here. For details, see [59,60,61,62,63].

#### 2.2. The Friedrichs Model

- (i)
- $|1\rangle $ is a Eigenvector of ${H}_{0}$ with Eigenvalue ${\omega}_{0}>0$. This Eigenvector belongs to the Hilbert space domain of ${H}_{0}$ and, therefore, it represents a bound state, ${H}_{0}\phantom{\rule{0.166667em}{0ex}}|1\rangle ={\omega}_{0}\phantom{\rule{0.166667em}{0ex}}|1\rangle $.
- (ii)
- Each of the $|\omega \rangle $ is an Eigenvector of ${H}_{0}$ with Eigenvalue $\omega \in {\mathbb{R}}^{+}$, ${H}_{0}\phantom{\rule{0.166667em}{0ex}}|\omega \rangle =\omega \phantom{\rule{0.166667em}{0ex}}|\omega \rangle $, the absolutely continuous spectrum of ${H}_{0}$. These Eigenvectors do not belong to the Hilbert space, but to the antidual of a RHS $\mathrm{\Phi}\subset \mathcal{H}\subset {\mathrm{\Phi}}^{\times}$, with $\mathcal{H}$ the Hilbert space where ${H}_{0}$ and H act. The details of the construction of this RHS may be seen in [76].

## 3. Gamow States as Functionals on an Algebra of Operators

#### 3.1. Functionals over the Algebras

- (i)
- They are differentiable at all points and at all orders.
- (ii)
- The value of these functions and all their derivatives at the origin is zero.
- (iii)
- They and all their derivatives go to zero at the infinity faster than the inverse of any polynomial.

- (i)
- The functions in both ${\Xi}^{\pm}$ are Schwartz functions, due to the properties of the Fourier transform [20].
- (ii)

#### 3.2. The Gamow Functionals

- (i)
- f is positive, i.e., for any $O\in \mathcal{A}$, one has that $f({O}^{\u2020}\phantom{\rule{0.166667em}{0ex}}O)\ge 0$.
- (ii)
- f is normalized, i.e., $f(I)=1$, where I is the identity in $\mathcal{A}$.
- (iii)
- If $\mathcal{A}$ were endowed with a topology compatible with the algebraic structure, f should be continuous with respect to this topology and the usual topology on the complex plane $\mathbb{C}$.

- (i)
- Pure states:If there exists a square integrable function $\psi (E)\in {L}^{2}({\mathbb{R}}^{+})$ such that$$\rho (E)={|\psi (E)|}^{2}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}\rho (E,{E}^{\prime})={\psi}^{\ast}(E)\phantom{\rule{0.166667em}{0ex}}\psi ({E}^{\prime})\phantom{\rule{0.166667em}{0ex}}.$$Note that $\rho (E,E)=\rho (E)$.
- (ii)
- Mixtures:Just defined by the relation $\rho (E)=\rho (E,E)$. Note that pure states are a particular case of mixtures. For mixtures we do not need the existence of a square integrable function satisfying (75).
- (iii)
- Generalized states:All the others. This include the van Hove states with diagonal singular [79,80,81], characterized by $\rho (E)\ne \rho (E,E)$, where they are still regular functions. Here, we may include states where one or both of the functions $\rho (E)$ or $\rho (E,{E}^{\prime})$ are generalized functions (distributions).

## 4. Coherent Gamow States

#### Resolutions of the Identity

## 5. From Non-Commutativity to Commutativity

## 6. Irreversible Phenomena and Loschmidt Echo

- (i)
- A quantum system with an initial state $|\psi ({t}_{0})\rangle $ is prepared at time ${t}_{0}=0$.
- (ii)
- During a interval of time $\tau $ a magnetic field B is applied in such a way that the system evolves according to the evolution operator $U(\tau )={e}^{-iH\tau}$. During this step the system is said to move “forward in time.”
- (iii)
- Then, during the same interval of time $\tau $, the magnetic field is reversed in such a way that the system evolves according to the evolution operator $U(-\tau )={e}^{iH\tau}$. During this step the system is said to move “backwards in time".
- (iv)
- The magnetic field is turned off and the initial state $|\psi ({t}_{0})\rangle $ and the final state $|\psi ({t}_{f})\rangle $, with ${t}_{f}=2\tau $, are compared.

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. A Model of Krein Space for Gamow Vectors

- (i)
- The functions (A4) are square integrable and, therefore, belong to ${L}^{2}(\mathbb{R})$.
- (ii)
- These functions do not belong to the domain of the multiplication operator as $\frac{E}{E-{z}_{R}}$ and $\frac{E}{E-{z}_{R}^{\ast}}$ are not square integrable on $\mathbb{R}$. Note that the multiplication operator on the energy representation is the Hamiltonian, under certain conditions on it [52,53] that we may assume without loosing nothing essential. Thus, the functions (A4) are out of the domain of the Hamiltonian. In addition, as one may directly check from the definition of Hardy function on a half plane, we have that $\frac{1}{E-{z}_{R}}\in {\mathcal{H}}_{+}^{2}$ and $\frac{1}{E-{z}_{R}^{\ast}}\in {\mathcal{H}}_{-}^{2}$.

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**MDPI and ACS Style**

Gadella, M.; Fortín, S.; Jorge, J.P.; Losada, M.
Mathematical Models for Unstable Quantum Systems and Gamow States. *Entropy* **2022**, *24*, 804.
https://doi.org/10.3390/e24060804

**AMA Style**

Gadella M, Fortín S, Jorge JP, Losada M.
Mathematical Models for Unstable Quantum Systems and Gamow States. *Entropy*. 2022; 24(6):804.
https://doi.org/10.3390/e24060804

**Chicago/Turabian Style**

Gadella, Manuel, Sebastián Fortín, Juan Pablo Jorge, and Marcelo Losada.
2022. "Mathematical Models for Unstable Quantum Systems and Gamow States" *Entropy* 24, no. 6: 804.
https://doi.org/10.3390/e24060804