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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