PHYSICAL REVIEW A VOLUME 11, NUMBER FEBRUARY 1975 Bound states in the continuum Frank H. Stillinger and David R. Herrick Bell Laboratories, Murray Hill, New Jersey 07974 (Received 5 November 1974) Quantum-mechanical examples have been constructed of local potentials with bound eigenstates embedded in the dense continuum of scattering states. The method employed corrects and extends a procedure invented by von Neumann and Wigner. Cases are cited whereby deformation of the local potential causes the continuum bound state to move downward through the bottom of the continuum, and to connect analytically to a nodeless ground state. A doubly excited model atom is also displayed, with interactions between its two "electrons, " having an infinite lifetime (in the Schrodinger equation regime). In the light of these examples, attention is focused on quantitative interpretation of real tun- neling phenomena, and on the existence of continuum bound states in atoms and molecules. I. INTRODUCTION that parameter in the neighborhood of its value corresponding to the crossing. In 1929, von Neumann and Wigner' claimed that Several generalizations are examined in Sec. IV. the single-particle Schrodinger equation could Included are cases involving higher angular mo- possess isolated eigenvalues embedded in the con- mentum, dimensionality differing from three, tinuum of positive energy states. They offered a Coulomb interactions, and external electric fields. constructive method, based upon amplitude modu- In each of these categories, families of localized lation of a free-particle wave function, leading to steady-state solutions to the Schrodinger equation a localized (i.e., integrable) eigenfunction and a can be constructed, with energies embedded in the local potential which produces it. The potential continuum. was bounded and could be made to vanish at in- In order to pave the way for eventual considera- finity. Diffractive interference was cited as the tion of many-electron systems, we have devised reason such localized positive-energy states could a model involving a doubly excited atomic state. exist. The specific two-electron realization appears in The von Neumann-Wigner states have gained Sec. V. Despite interactions between the "elec- importance recently with the growing suspicion trons" which one might expect to cause autoioniza- that some atomic and molecular systems might tion, the state lives forever (in the Schrodinger also exhibit bound states in the relevant continua, ' ' description). For that reason, we have reexamined the von Neu- We terminate this exploratory paper with dis- mann-Wigner analysis with a view to exposing cussion of a wide variety of chemical and physical those directions in which it could fruitfully be phenomena whose quantitative interpretation may generalized. The results obtained thus far raise require careful accounting for the possible exis- interesting questions about the validity of conven- tence of bound states in the continuum. tional theory for tunneling phenomena, and for certain resonance processes in atomic and molec- ular spectroscopy. II. VON NEUMANN-WIGNER METHOD The original von Neumann-Wigner paper con- Using suitable reduced units, we consider the tains a superficial algebraic error which affects single-particle wave equation its results in a significant way. However, that lapse in no way compromises the cleverness of (--,'V'+ V)e =re (2.1) those authors's underlying strategy. We provide to be solved in infinite three-space. We restrict the corrected version in the following Sec. II, for attention initially to potentials V which are bound- completeness. ed, and which are local operators in position rep- Section III demonstrates by concrete example resentation. Since an equivalent form of (2. is how localized states can be adiabatically manip- 1) ulated so that their energy drops below the bottom V =E+-,'[(V'4) 4/], (2 2) of the continuum. This process can occur for- it is obvious that nodes of the wave function 4' mally as the result of continuous change in a pa- must be matched by vanishing of its I aplacian. rameter in the potential. It is noteworthy that The free-particle S wave the energy eigenvalue, the potential, and the local- ized wave function can all be analytic functions of 4', (r) = (sinkr)/kr (2.3) 11 BOUND STATE S IN THE CONTINUUM satisfies Eq. (2.2) with energy eigenvalue approach cannot supply any other exact wave func- tions and their energies for that same potential. (2.4) But since V(r) vanishes at infinity, scattering and V identically zero. Although +0 is not in- states for all energies E) 0 will exist with wave tegrable, von Neumann and Wigner suggested that functions that could numerically be approximated it would be possible to generate an integrable wave by standard techniques. The existence of the function by modulating the amplitude of 4'0 in an bound state in this continuum requires a discon- appropriate way. If one writes tinuous increase by m in the S-wave scattering the incident rises through e(r) =+, phase shift as energy (r)f(r), the fixed value 2&'. In other words, the bound then 4' integrability only requires that f (r) drop state amounts to an infinitely narrow resonance. ' to zero as r, P ) —,', for diverging ~. A far wider class of modulating functions f can Substituting expression (2.5) into Eq. (2.2), one be considered, such as obtains f(r) = (A'i[2kr —sin(2kr)] "j ", ——,'k'+ V(r ) = E k(cotkr) f'(r)/f (r) m- -'„mn)-,', (2.12) (2.6) for which a corresponding V(r) can be derived. We note in passing that von Neumann and Wigner Note, however, that a choice which decays ex- erroneously obtained "tan" in place of "cot" for ponentially to zero as &-~ is not consistent with their version of this last equation. ' vanishing of V(r) in the same limit. It should also In order that V(r) remain bounded, it is clear be stressed that no f(r) seems to exist, producing from Eq. (2.6) that f'/f must vanish at the poles a bound state in the continuum, whose V(r) decays of cot(kr), i.e. at the zeros of sin(kr). This can more rapidly with increasing & than the type of , ' be achieved by selecting f (r) to be a differentiable modulated r shown in Eq. (2.11). function of the variable r III. ANALYTIC CONTINUATION ACROSS THE dr' = —,'kr ——,' sin'(kr') sin(2kr). (2.7) CONTINUUM EDGE 0 the positive-energy The specific choice suggested by Ref. 1 is The eigenvalue for preceding bound state can be moved up or down within the f (r) = (A'+ [2kr —sin(2kr)] 'j ', (2.8) continuum merely by varying k. Equation (2.9) specifies the way in which V(r) must deform to where A. is an arbitrary nonzero constant. Be- continue supporting its bound state. We shall now cause decreases as r ' in the limit r-~, it f (r) see that it is pcssible analytically to carry such a is obvious that 4' is integrable. bound state downward, through the bottom of the Having made assumption (2.8) about the form of continuum, into the negative energy regime. can substitution that the terms f, one verify by This feature can easily be demonstrated for the containing in (2.6) vanish as r-~. There- f Eq. corrected von Neumann-Wigner example. How- fore itself will vanish in this limit provided V(r) ever, it turns out that after the crossing the po- that E continues to be identified with 2O'. The tential V(r) approaches an infinite-r limit that be- potential then can be readily derived: gins to rise above zero. The continuum edge 64&'A' s in4ks would of course have to rise as well. Fortunately, [A.' y (2kr —sin2kr)'] ' we can circumvent this peculiarity by examining instead any of several slightly modified examples 48k' sin'kr —Sk'(2kr —sin2kr) sin2kr for which V(r-~) remains zero after the crossing. A'+ (2kr —sin2kr)' One possible modification consists first in re- placing the variable shown in Eq. (2.7) by the (2 9) equally acceptable choice Near the origin, the function behaves thusly: s(r) = 8k2 ', r' sin~(kr') dr' V(r) = (80/SA' —64)k'(kr)4+P((kr)'), (2.10) ' while its large-& character is' =-, (2kr)' —2kr sin(2kr) —cos (2kr) + 1. (3.1) V(r) - -Sk'(sin2kr)/2kr. (2.11) If the modulating factor f is now taken to be a func- tion of s, the poles of cot(kr) in Eq. (2.6) will con- have established that the local potential (2.9) We tinue to be cancelled. For simplicity we use possesses a bound (that is, normalizable) eigen- state with positive energy 2O'. Unfortunately, the (3.2) FRANK H. STILLINGER AND DAVID R. HEBRICK where again A is nominally an arbitrary param- 0 (r, A. & &,) = (s inh sr)/sr[a' g' + s (x)] eter. Since & rises quadratically with & in the ~a~2 K r )- t large-r regime, wave-function normalizability is (2 ~ (3.11) assured. By using expression (3.2} in Eq. (2.6), The potential becomes the potential which produces the bound S state, 64«' sinh'x& 4, is found to be f, [a'8 + s (r)]' 64k'r' sin'kx 4k'(sin'kr+2kr sin2kr) [A'+ s(r)]' A'+ 4 z'(s inh'xr + 2 mr sinh2 vx) s(r) g'z'+s(r) (3.3) somewhat surprisingly, the long-range behavior of This modified potential displays the same asymp- I/ has become Coulombic: totic form shown in Eq. (2.11)for the prior case. However, near the origin V(~, a&A.,)-z/r, ~&0. (3.13) V(r) = -(20k'/&')(k~)'+0((k&)'}, (3.4) The energy E is trivially analytic in ~ at ~„by definition.