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What is a resonance?. KEK Lecture (1). K. Kato Hokkaido University Oct. 6, 2010. ( 1 )  What is a resonance ?. The discrete energy state created in the continuum energy region by the interaction, which has an outgoing boundary condition. However, there are several definitions of resonances.

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What is a resonance
What is a resonance?

KEK Lecture (1)

K. Kato

Hokkaido University

Oct. 6, 2010

What is a resonance

1) What is a resonance?

The discrete energy state created in the continuum energy region by the interaction, which has an outgoing boundary condition.

However, there are severaldefinitions of resonances

What is a resonance

(i) Resonance cross section

s(E) ~ —————

Breit-Wigner formula

(E – Er)2 + Γ2/4

What is a resonance

(ii) Phase shift

“Quantum Mechanics” by L.I. Schiff

… If any one of klis such that the denominator ( f(kl) ) of the expression for tanl,

|tanl| = | g(kl)/f(kl) | ∞ ,

( Sl(k) = e2il(k) ),

is very small, the l-th partial wave is said to be in resonance with the scattering potential.

Then, the resonance: l(k) = π/2 + n π

What is a resonance

What is a resonance

(iii) Decaying state

“Theoretical Nuclear Physics”by J.M. Blatt and V.F. Weisskopf

We obtain a quasi-stational state if we postulate that for r>Rc the solution consists of outgoing waves only. This is equivalent to the condition B=0 in

ψ (r) = A eikr + B e-ikr(for r >Rc).

This restriction again singles out certain define solutions which describe the “decaying states” and their eigenvalues.

What is a resonance

For the resonance momentum kr=κ–iγ,

ψ(r) = ei κr erγ, (not normalizable (γ>0))

What is a resonance

G. Gamow, Constitution of atomic nuclei and dioactivity (Oxford U.P., 1931)

A.F.J. Siegert, Phys. Rev. 56 (1939), 750.

The physical meaning of a complex energy

E=Er– iΓ/2

can be understood from the time depen-dence of the wave function

ψ(t) = ψ(t=0) exp(-iEt/h)

and its probability density

| ψ(t)|2 = |ψ(t=0)|2 exp(-Γt/2h).

The lifetime of the resonant state is given by τ = h/Γ.

What is a resonance

4. Poles of S-matrix

The solution φl(r) of the Schrödinger equation;

Satisfying the boundary conditions


the solution φl(r) is written as

What is a resonance

where Jost solutions f±(k, r) is difined as

and Jost functions f±(k)

Then the S-matrix is expressed as

The important properties of the Jost functions:



From these properties, we have unitarity of the S-matrix;

What is a resonance

  • Ref.

  • J. Humblet and L. Rosenfeld, Nucl. Phys. 26 (1961), 529-578

  • L. Rosenfeld, Nucl. Phys. 26 (1961), 594-607.

  • J. Humblet, Nucl. Phys. 31 (1962), 544-549.

  • J. Humblet, Nucl. Phys. 50 (1964), 1-16.

  • J. Humblet, Nucl. Phys. 57 (1964), 386-401.

  • J.P. Jeukenne, Nucl. Phys. 58 (1964), 1-9

  • J. Humblet, Nucl. Phys. A151 (1970), 225-242.

  • J. Humblet, Nucl. Phys. A187 (1972), 65-95.

What is a resonance

2) Many-body resonance states

  • (1)Two-body problems; easily solved

  • Single channel systems

  • Coupled-channel systems

  • (2) Three-body problems; Faddeev

  • A=C1+C2+C3

  • Decay channels of A

  • What is a resonance

    A[C1-C2]B+C3, Eth(C3)

    [C2-C3]B+C1, Eth(C1)

    [C3-C1]B+C2, Eth(C2)

    B [C1-C2]R+C3, Eth(C12)

    [C2-C3]R+C1, Eth(C23)

    [C3-C1]R+C2, Eth(C31)

    C C1+C2+C3, Eth(3)

    What is a resonance

    Multi-Riemann sheet

    Eth(C3)  Eth(C2) Eth(C2) Eth(3)





    (3) N-Body problem; more complex

    What is a resonance

    Eigenvalues of H(q) in the complex energy plane

    Complex scaling

    U(q) ; r rei q

    k ke-i q

    Yq= U(q) Y(r)

    =ei3/2 q Y(rei q)

    H(q)= U(q) H U(q)-1


    What is a resonance

    Complex Scaling Method

    physical picture of the complex scaling method

    Resonance state

    The resonance wave function behaves asymptotically as

    When the resonance energy is expressed as

    What is a resonance

    the corresponding momentum is

    and the asymptotic resonance wave function


    What is a resonance

    This asymptoticdivergence of the resonance wave function causes difficulties in the resonance calculations.

    In the method of complex scaling, a radialcoordinate r is transformed as

    Then the asymptotic form of the resonance wave function becomes


    What is a resonance

    It is now apparent that when π/2>(θ-θr)>0 the wave function converges asymptotically. This result leads to the conclusion that the resonance parameters (Er, Γ) can be obtained as an eigenvalue of a bound-state type wave function.

    This is an important reason why we use the complex scaling method.

    What is a resonance

    Eigenvalue Problem of the Complex Scaled Hamiltonian

    • Complex scaling transformation

    • Complex Scaled Schoedinger Equation

    What is a resonance

    • ABC Theorem

    • J.Aguilar and J. M. Combes; Commun. Math. Phys. 22 (1971), 269.

    • E. Balslev and J.M. Combes; Commun. Math. Phys. 22(1971), 280.

    • i) cq is an L2-class function:

    • ii) Eq is independent on q ( )

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