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Lectures on Physical andMathematical aspects ofGamow States

by Osvaldo Civitarese

from the Departmentof Physics. University of

La Plata. Argentina

ELAF 2010-Ciudad de Mexico


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Resonances can be defined in several ways,but it is generally accepted that, under

rather general conditions, we can associatea resonance to each of the pairs of poles ofthe analytic continuation of the S-matrix.


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In spite of the long-time elapsedsince the discovery of alpha decay phenomena in nucleiand their description in terms of resonances, the use of

the concept in nuclear structure calculations hasbeen hampered by an apparent contradiction withconventional quantum mechanics, the so-called

probability problem. It refers to the fact that a statewith complex energy cannot be the eigenstateof a self-adjoint operator, like the Hamiltonian,

therefore resonances are not vectors belonging tothe conventional Hilbert space.


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These lectures are devoted to the

description of resonances, i.e. Gamow states,in an amenable mathematicalformalism, i.e. Rigged Hilbert Spaces. Since

we aim at further applications in the domainof nuclear structure and nuclear many-bodyproblems, we shall address the issue in aphysical oriented way, restricting thediscussion of mathematical concepts to the

needed, unavoidable, background.


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From a rigorous historical prospective, thesequence of events and papers leading to ourmodern view of Gamow vectors in nuclearstructure physics includes the following

steps:


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a) The use of Gamow states in conventionalscattering and nuclear structure problemswas advocated by Berggren, at Lund, in the

1960s, and by Romo, Gyarmati and Vertse, andby G. Garcia Calderon and Peierls in the late70s. The notion was recovered years later, byLiotta, at Stockholm, in the 1980s, inconnection with the microscopic description of

nuclear giant resonances, alphadecay and cluster formation in nuclei. Both atGANIL-Oak Ridge, and at Stockholm, the use

of resonant states in shell model calculationswas (and still is) reported actively.


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b) A parallel mathematical developmenttook place, this time along the lineproposed by Bohm, Gadella and

collaborators.

Curiously enough both attempts went

practically unnoticed to each other forquite a long time until recently, when someof the mathematical and physical

difficulties found in numerical applicationsof Gamow states were discussed oncommon grounds ( Gadella and myself).


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Technically speaking, one may summarizethe pros and drawbacks ofBerggren andBohm approaches in the following:

(i)The formalism developed by Berggren isoriented, primarily, to the use of a mixedrepresentation where scattering states

and bound states are treated on a foot ofequality. In his approach a basis shouldcontain bound states and few resonances,

namely those which have a small imaginarypart of the energy.


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ii)In Bohms approach the steps

towards the description of Gamowstates are based on thefact that the spectrum of the

analytically continued Hamiltoniancovers the full real axis. As adifference with respect to

Berggrens method, the calculationsin Bohms approach are facilitatedby the use of analytic functions.


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The subjects included in these lectures

can be ordered in two well-separatedconceptual regions, namely:(a) the S-matrix theory, with reference

to Moller wave operators, to the spectraltheorem of Gelfand and Maurin, and tothe basic notions about Rigged Hilbert

Spaces, and(b) the physical meaning of Gamowresonances in dealing with calculation of

observables.


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The mathematical tools needed tocover part (a) are presented in a self -contained fashion.

In that part of the lectureswe shall follow, as closely as possible,

the discussion advanced in the work ofArno Bohm.


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Gamow vectors and decaying

states Complex eigenvalues

A model for decaying states An explanation and a remedy for the

exponential catastrophe Gamow vectors and Breit-Wigner

distributions


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Some references

A. Bohm, M. Gadella, G. B. Mainland ;Am.J .Phys 57(1989)1103

D. Onley and A. Kumar ;Am.J .Phys.60(1992)432 H. J akobovits, Y. Rosthchild, J . Levitan ; Am.J . Phys.63

(1995)439)

H. Massmann, Am. J . Phys. 53 (1985) 679 B. Holstein; Am.J . Phys. 64 (1996) 1061

G. Kalbermann; Phys. Rev. C 77 (2008)(R)041601

G. Garcia-Calderon and R. Peierls; Nuc.Phys.A265(1976)443.

T. Berggren; Nuc. Phys. A. 109 (1968)265.


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-Decay law

Number of particles that have not decay at the time t

(initial decay rate)

(survival)


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For a vector that represents thedecaying state, like

The probability of survival is written

Therefore, this absolute value should beequal to the exponent of the decay rate


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The decay condition can be achieved if we think ofthe state as belonging to a set of states withcomplex eigenvalues

From where it follows that


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The equality between the experimental andtheoretical expressions for the probabilityof survival imply

which is, of course, a rather nice but strangerelationship which has been obtained under theassumption that there are states, upon which weare acting with of a self-adjoint operator (theHamiltonian), which possesses complex eigenvalues.


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A model for decaying systems

A particle of point-mass (m)

confined in a sphericallysymmetric potential well

The solutions of the non-relativisticSchroedinger equation are written (zero angular

momentum for simplicity)


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The stationary solutions are obtained in the

limit of a barrier of infinite height, andthey are of the form (inside the barrier)

with the real wave number


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And for the three regions considered thequasi-stationary solutions are of the form

the wave-number are real (k) inside andoutside the barrier region and complex (K)inside


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By requiring the continuity of the function and its

first derivative, and by assuming purely outgoingboundary conditions to the solution in the externalregion, one gets


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The solution of the system of equationsyields

and the relation between k and K (eigenvalueequation)


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The initial decay rate can be calculated from

the flux of the probability current integratedover the sphere

where

(radial component of the current in the externalregion)


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After some trivial calculations one gets (for

the external component of the solution)

Since the real part of the energy is much

smaller that the barrier (but positive) we have


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Then, by replacing G for its value in terms

of the approximated wave numbers, we find


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and

If we proceed in the same fashionstarting from the definition of the Gamowvector, we find for the complex wave

number


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by keeping only leading order terms, sincethe imaginary part of the energy is muchsmaller that the real part:


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With these approximations, the relationship between

k and K determines the structure of the decay width:

and


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And by equating the imaginary parts of both

equations one gets


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which is fulfilled if (at the same

order)

This last result is quite remarkable,because it shows that decaying states can

indeed be represented by states withcomplex eigenvalues. It also leads to therather natural interpretation of the decay

time as the inverse of the imaginary partof the complex energy.


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The exponential catastrophe

For well behaved vectors in Hilbert space

one has the probability density (finite andnormalizable)

But not all vectors are proper vectors,since (for instance for Diracs kets)


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And it means that, for them

Although it is finite at each point (in space) it isinfinite when integrated over the complete domain.For a Gamow vector one has, similarly:


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from the wave function in the externalregion one has

one has


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The factor between parenthesis can be writen in

terms of a initial time associated to theapproximated wave-number, leading to:

This result (which is valid for times t greater than theinitial time) means that a detector place at adistance r detects a counting rate that is maximum atthe initial time and decreases exponentially withincreasing time (Bohm-Gadella-Mainland)


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Thus, when correctly interpreted,Gamow vectors leads to anexponential decay law and not toan exponential catastrophe.


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Gamow vectors and Breit-Wigner distributions

At this point (and as a first contact withthe proper mathematics) we shallintroduce another tool to understandGamow vectors. That is the analogous ofDiracs spectral theorem and the nuclear

spectral theorem (also called Maurin-Gelfand-Vilenkin theorem). This will beour first exposure to the concept of

Rigged Hilbert Spaces (or Gelfandstriads).


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Dirac spectral theorem

In the infinite dimensional Hilbert space we candefine an orthonormal system of eigenvectors of

an operator with a discrete set of eigenvalues(the Hamiltonian operator, for instance), suchthat

Then, every vector of the same

space can be expanded as:


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In scattering problems the Hamiltonian has a continuous

spectrum (in addition to bound states), and this part ofthe spectrum can be added to the expansion

To show that this is indeed a correct expansion we shall introduce,in addition to the Hilbert space (of normalizable, integrablefunctions) a subspace of all integrable functions in the energy

representation. We think of these functions as well behaved(infinitely differentiable and rapidly decreasing in the energy rep)(technically speaking they are elements of the Schwartz space).


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We also need another space, which should

accomodate the antilinear mappings of thesefunctions (something is antilinear ifF(ax+by)=a*F(x)+b*F(y)).

We then have:

a)The Hilbert space of regular vectors

b)The space (Schawrtz) of test functions

c)Its antilinear mapping.Under these conditions, Gelfand defines thetriads (or Rigged Hilbert Spaces, or triplets) as

the construction


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Nuclear spectral theorem:

Let H be a self-adjoint operator in the Hilbertspace. Then it exists in the subspace

a set of elements with the property

Such that any function of (b) can be

written in the form


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Let us see how these notions apply in the case ofGamow vectors

A particular generalized eigenvector with complexeigenvalues can be defined by:

which is a functional over well behaved functions


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Properties and conditions on these vectors:

1) (Generalized eigenvalue equation)

2) (Time evolution)

(Probability of detecting a givenvalue of the energy E)

3)

Proof of the properties


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p p

(use Cauchy integral formula):


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(use Cauchy integral formula):

The integral is calculated over a closed contour(real energy axis) and an infinite semicircle in thelower half of the complex plane, leading to


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since

vanishes on the infinite semicircle in thelower half of the complex plane only fort >0.

More generally, since there is one (pair of ) S-matrix pole at E,then every well behaved vector can be given by the generalizedexpansion ( A. Mondragon has shown it first, and then with E.

Hernandez) we shall come back to this result later on)

The approximation to the exponential decay


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pp p ylaw

The exponential decay law is aconsequence of the choice of the Breit-

Wigner energy distribution (a naturalchoice, indeed).

Influence of the direct integration overthe physically positive energyspectrum: it affects the decay law


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Then, if we integrate a givenamplitude, like

in the real axis, and use Cauchy formula, we get,

for the relevant part of the integral

that the main contribution comes from the pole locatedat the lower half-plane, at it yields:


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where the functions K are modified


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where the functions K are modifiedBessel functions of imaginary argument.The term by term expansion of A(t)reads


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Then, the first term already gives the expecteddependence (decay law) as a function of theimaginary part of the complex energy, and it isvalid provided t>>1/Real part of (E).


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then

and

from which


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from which

d i h h difi d B l f i


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and with the modified Bessel functions


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Gamow vectors: a tour from the


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Gamow vectors: a tour from the

elementary concepts to the (moreelaborate) mathematical concepts

Some references:

R de la Madrid and M. Gadella ; Am. J. Phys.70 (2002) 626

R de la Madrid, A. Bohm, M. Gadella ;

Forstchr. Phys. 50 (2002) 185O. Civitarese and M. Gadella; Phys. Report.396 (2004) 41

-We shall compare the description of Gamowt s


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vectors

1)in the scattering problem,

2)the S-matrix formalism, and

3)in the rigged Hilbert space framework

All are equivalent way of introducing Gamowvectors and their properties, but some are moreillustrative than others, as we shall see in the

following notes.

-Resonances for a square barrier potential


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Resonances for a square barrier potential

Since we have already discussed the case in thefirst lecture, we shall review here the main results andexpressions for a later use. We shall start with

Schroedinger equation in the Dirac notation. When theenergies belong to the continuum spectrum of H, thevectors are Diracs kets and they are not vectors in theHilbert space (they are not square integrable). However,we can use them to expand the space of wave functions


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We are searching for solutions ofthe equation

in spherical coordinates it reads


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with


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The solutions are

Boundary and continuity conditions


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they are of the form


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they are of the form

with solutions


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Thus, the zero angular momentum solution reads


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Thus, the zero angular momentum solution reads

with

the expansion of the wave function will then be


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the expansion of the wave function will then be

and from it one has (in bra and kets notation)

S matrix


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S-matrix

the probability-amplitud of detecting an out-state


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(superscipt -) in an in-state (superscript +) is


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the S-matrix, in the energy representation, iswritten

the continuation of S is analytic except at its

poles, which are determined by


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that is

the solutions come in pairs of complex

conjugate values of the energy,

Decaying resonance energies (square well potential)


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Growing resonance energies (square well potential)

-resonance wave numbers (for the square wellpotential)


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Gamow vectors


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The time independent Schroedinger equation iswritten

and the radial part of the zero angular momentumGamow vector is given by


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with the complex wave number

and


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the boundary conditions are:

notice the purely outgoing conditionat infinity


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the purerly outgoing boundary condition(impossed to Gamow vectors) is often written

next, we have to solve the equations to calculate theradial wave functions explicitely, in matrix notationsthey read


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where

the determinant of the system is then written


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leading to the dispersion relation

Then for each pair of complex energies we findthe decaying


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and growing radial solutions


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Green function method

the formal solutions are

h h f h d f f


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where the functions entering the definition ofG are the solutions of the radial equation whichvanishes at the origin

or fulfills a purely outgoing boundary condition

and their wronskian


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Th b d diti f th t i


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The boundary conditions for the outgoingwave are

With l ti i b


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With solutions given by

the wronskian is of the form

the structure of the Green function is,

th f


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therefore

With residues at the poles (which

are the same poles of S)

Complex vector expansions


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From the amplitude

We can extract the contribution of resonances bymaking an analytic continuation of the S-matrix

and by deforming the contour of integration


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From it, one gets


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and a similar expression holds for the out-state


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Contribution from decaying Gamow vectors


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Contribution from growing Gamowvectors

Time asymmetry of the purely outgoing

boundary condition:


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boundary condition:for kd=Re(k)-iIm(k); with Re(k), Im(k)>0 (fourthquadrant) we write

and

In the same fashion, for the growing vectorswe have for kg=-Re(k)-iIm(k); (with Re(k),Im(k)>0, third quadrant )


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and

Leading to the probability densities


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and


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Rigged Hilbert Space treatment of continuumspectrum


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Mathematical concepts

Observables

Reference: O. Civitarese and M. Gadella

Phys. Rep. 396 (2004) 41, and references quotedtherein.

The Hilbert space of scatteringstates


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We can write the Hilbert space of a self-adjoint (central)Hamiltonian as the direct orthogonal sum of the discrete (d)and continuous (c) subspaces

And, moreover, we can decompose the continuoussubspace into two mutually orthogonal parts (absolutelycontinuous (ac) and singular continuous (sc) parts)

Scattering states are not-bound regular states andthey belong to to the ac part of the spectrum, otherstates are contained in sc (fractal section of the


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(spectrum of H), thus

The Moller operator


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M pWe assume that any scattering state isasymptotically free in the past. For any scatteringstate , it exists a free state such

As the limit in a Hilbert space is takenwith respect to its norm, this is equivalent to say that

Since the evolution operator is unitary,we have:


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p y,we have:

Then, we can de


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These operators are the Moller wave operators


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p p

with the properties

Proof:


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With this choice


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Which is equivalent, in infinitesimalform, to

Analytic continuation of S(E)->S(p=+2mE)(complex p-plane or k-plane)

(i) Single poles in the positive imaginary axis of S(z)that correspond to the bound states of H.


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that correspond to the bound states of H.(ii) Single poles in the negative imaginary axis thatcorrespond to virtual states.(iii) Pairs of poles, in principle of any order, in thelower half-plane. Each of the poles ofa pairhas the same negative imaginary part and the same

real part with opposite sign.Thus, if p is one of these two poles the other is p*(complex conjugation). These poles are

called resonance poles and in general there is aninfinite number ofthem


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We start with a definition of RHS:

A t i l t f i i d Hilb t (RHS)


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A triplet of spaces is a rigged Hilbert space (RHS)if:

The intermediate space H is an infinite-dimensional Hilbert space.

Is a topological vector space, which is densein H (we call this space left-sail space ofRHS)

Is the anti-dual space of

(we call this space : right-sail

space of RHS)

RHS are useful, among other applications, for:

1. Giving a rigorous meaning to the Diracformulation of quantum mechanics .


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In this case,it is customary to demand that the

vector space on the left be nuclear(e.g.; obey theNuclear Spectral Theorem, according to which

every observable, or set of commuting

observables, has a complete set of generalizedeigenvectors whose corresponding eigenvaluesexhaust the whole spectrum of the

observable. This is in agreement with the Diracrequirement


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Rigged Hilbert spaces have the followingproperties:

Property 1. Let A be an operator on H. If we defineh d i (D) f i dj i h h f ll i


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the domain (D) of its adjoint, then the followingconditions are fulfilled:

(i) The domain contains the left-sail space of theRHS.(ii) For each function belonging to the left-sail

space the action of the adjoint operator on it alsobelongs to the left-sail space.(iii) The adjoint operator is continuous on the left-

sail space . Then


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The mean values are zero

If we define the action of the Hamiltonian on the Gamowvector as:


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Comment: the object

Is not defined, since the vectors do notbelong to the Hilbert space of H.


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The mean values are real

For a mapping of analytic functions (like projections on thepositive real semi-axis) the norms are finite,


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positive real semi axis) the norms are finite,


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In momentumspace (Berggren) one has


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This is Berggrens main result.


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and


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In Berggrens formalism


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and several other references fromother authors and from myself too


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other authors and from myself toolazy to compile them here, sorryIwill quoted them in the written

version of the lectures.


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For a finite value of


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Graph of the ratio R, as

a function of the energy


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Alpha decay revisited(ref: B. Holstein; Am.J.Phys. 64(1996) 1061)

The alpha-decay rate is expressed


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where is the WKB barrier penetration factor

/2 is the frequency at which the pre-formed alpha-particle strikes the barrier,and is a factor of the order of unity.


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The radial wave functions are of the form


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but, simultaneously

Which implies that the energies which

may fulfill it must be complex The


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may fulfill it must be complex. Theassociated dispersion relation

since the right hand side is suppressed, this

equation yields

This is exactly the condition found in the first

lecture, in dealing with the definition of Gamowvectors for a single barrier.

Following the conventional WKB treatment


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Following the conventional WKB treatment(see Holstein) we define the probabilityflux and the decay rate as


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Naturally, we can understand this result by

explicitely writting for the complex eigenvalue

in


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in


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In order to describe resonances,we consider the reduced resolvent of H in the bound state


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which is given by


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The form of the analytic continuations is


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The poles are determined by thezeros of the equations


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Here, we intend to get the explicit

form of the Gamow vectors forthe Friedrichs model.

Let x be an arbitrary positivenumber (x>0) and write the


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number (x 0) and write theeigenvalue equation

with

By applying to it the complete Hamiltonian we get thesystem of equations


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It yields an integral equation with one solution of

of the form

This is a functional in the right-sailspace of the RHS. It admits acontinuation to the lower half plane,

with a single pole:


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then


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thus

This is the formal solution of the decayingGamow vector for the Friedrichs model


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Gamow vector for the Friedrichs model .In the same way, we have for the growing

Gamow vector the expression


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With the quantities A, B, etc, givenby


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The equation of motion (in the

interaction picture)

ld h f h l d


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Yields the equations for the amplitudes

The, from both equations one gets

After a long time (t->) b(k,) is theprobability amplitude to find the boson state


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p y pand a() is the survival amplitude in the p-h

state. If a(t


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therefore

This integral is transformed as an


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This integral is transformed as anintegral on complex energies

In the complex energy plane theintegration path is of the form


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Since now we have an explicitexpression for f() we can write, forthe survival amplitude


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Which may be solved, by integration, foran explicit structure of Z()


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And with it, we integrate the expressionof the survival amplitude in the regions

limited by the contour


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The contribution from the cut is a slowly varying


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The contribution from the cut is a slowly varying

function of the form


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The decay is exponential up to 5lifetimes (the time is given in units of

the lifetime), but the regime changes atabout 6 lifetimes due to interferencesbetween the pole and the cut


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Dependence of the survival uponthe coupling constant


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Onset of non-exponentialdecay, for the strongcoupling case


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Line shapes (for the continuouspart of the spectrum, for w_01000 MeV

Assisted tunneling of a metastable statebetween barriers (G. Kalbermann; Phys. Rev.

C. 77 (2008) 041601 (R)

Problem: exposing a system to external

excitations can enhance its tunneling rate.


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Model:

Initial non-stationary state

Stationary states (even and odd solutions)


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Normalization factors (evensolutions)


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Normalization factors (odd solutions)


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External perturbation

Wave function


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Amplitudes:


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Matrix element of the external potential


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Then, the eqs. for the amplitudesread

(for even solutions)


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(for odd solutions)

Rescaling the variables:

The amplitudes are given by


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with these expressions the poles are locatedsymmetrically above and below the real momentumaxis, with values given by


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In Kalbermanns calculations, the polestructure is of the form:


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The first pole dominates the structure ofthe wave function, as of the decay constant,

which is then given by (even states)


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And (odd states)

Then, from Kalbermanns results it is seen that two

main effects are due to the action of theexternal potential upon the decaying states,namely:

a) The decay constant is enhanced

b) The dominance of the first pole produces an


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extra dampingThat is: the external field causes the acceleration

of the tunneling (the wave function will tunnelfaster)

Two-Particle Resonant States in a Many-BodyMean Field (from PRL 89 (2002) 042501

The role played by single-particle resonances and ofthe continuum itself upon particles moving in the continuum

of a heavy nucleus is not fully understood.One may approach this problem by using, asthe single-particle representation, theBerggren representation.


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One chooses the proper continuumas a given contour in the complex energy planeand forms the basis set of states as the bound states plusthe Gamow resonances included in that contour plus the

scattering states on the contour (quoted from thepaper)

One maythus think that the Berggren representation can also beused straightaway to evaluate many-particle quantities, asone does with the shell model using bound representations.Unfortunately this is not the case. The root of the problemis that the set of energies of the two-particle basis statesmay cover the whole complex energy plane of interest.


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Then, the main contribution to the imaginary part ofthe energy of the many-body states is given by

Gamow vectors in the single-particle basis, asexpected, with no effects coming from scatteringstates. To illustrate this point we go to the nexexample (dispersion of neutrinos by a nucleus)


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Some applications to nuclear structureand nuclear reactions

refs: O.C, R. J. Liotta and M. Mosquera(Phys. Rev. C. 78 (2008) 064308)

We study the

process (chargecurrent neutrino-nucleos scattering)


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By assuming that the single particle states to beused in the calculations belong to a basis with bound,quasi-bound, resonant and scattering states.

The proton-particle states are (al largedistances they behave as exp{ikr}):

a) bound states, for which Re(k)=0,Im(k)>0,

b)anti-bound states, for which Re(k)=0,Im(k)


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c) outgoing (decay) states for whichRe(k)>0, Im(k)


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The results of the calculations, show that:

1. the largest contributions to the consideredchannels of the cross section are given bynuclear excitations where bound and resonant

states participate as proton single-particlestates, and

2. the contribution of single-particle states in


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the continuum is, for all practical purposes,negligible.

Some final words after theseexamples.


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Acknowledgements

I want to express my gratitude toour colleagues, who havecontributed to the development

of the field and from whom I havehad the privilege of learning thetechniques presented in these


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lectures.

Since these is a very long list, indeed, I

warmly and gratefully thanks all ofthem in the persons of Profs. A. Bohm,M. Gadella, and R.J. Liotta


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Within few months three of the most outstandingphysicist of the twenty century: Aage Bohr (NBI),Carlos Guido Bollini (UNLP) and Marcos Moshinsky

(UNAM) passed away.

I have had the privilege of knowing them. They had

in common the joy of thinking physics and thegenerosity of sharing their feelings with theirstudents and colleagues .

I h h h i j i d


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I hope that their trajectories andmemories will remain with us and with the youngergenerations of physicists in the years to come.


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