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New families of isospectral Hamiltonians
Phys. Rev. D 33, 1048 – Published 15 February, 1986
DOI: https://doi.org/10.1103/PhysRevD.33.1048
Abstract
A new procedure is developed for generating families of Hamiltonians which share exactly the same set of eigenvalues. The new method is related to the Marchenko equation in much the same manner as the method of Abraham and Moses [Phys. Rev. A 22, 1333 (1980)] is related to the Gel’fand-Levitan equation. The two procedures in general yield inequivalent new families of Hamiltonians when used to insert or delete states, but are equivalent (with a proper choice of parameters) when used to renormalize a state. The effect of the new procedure on reflection and transmission amplitudes and on the norming constants for bound states is compared with corresponding results using the Abraham-Moses and Darboux techniques.
References (15)
- P. A. Deift, Duke Math. J. 45, 267 (1978). Deift introduced the term ``essentially isospectral'' for the relation between two operators whose sets of eigenvalues are identical except possibly for a zero eigenvalue. By a natural generalization, I use essentially isospectral to mean that the sets of eigenvalues are identical except possibly for a finite number of eigenvalues, and I use ``strictly isospectral'' or just ``isospectral'' to mean that the sets of eigenvalues are identical without exception.
- Note that two Hamiltonians may be isospectral without being ``spectrally equivalent,'' if the latter term is taken to imply that the two Hamiltonians have identical spectral density functions.
- See, for example, R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, New York, 1982), Chap. 20; Z. S. Agranovich and V. A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963); K. Chadan and P. C. Sabatier, Inverse Problems in Scattering Theory (Springer, New York, 1977).
- See, for example, G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980), Sec. 2.6.
- E. Witten, Nucl. Phys. B vec B 188, 513 (1981); P. Salomonson and J. W. Van Holten, ibid. B vec 196, 509 (1982); F. Cooper and B. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983); M. de Crombrugghe and V. Rittenberg, ibid. 151, 99 (1983); E. Gozzi, Phys. Lett. 129 B vec, 432 (1983).
- L. Gendenshtein, Pis'ma Zh. Eksp. Teor. Fiz. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; C. M. Bender, F. Cooper, and B. Freedman, Nucl. Phys. B vec B 219, 61 (1983); M. Bernstein and L. S. Brown, Phys. Rev. Lett. 52, 1933 (1984); V. Kostelecký and M. M. Nieto, ibid. 53, 2285 (1984); T. D. Imbo and U. P. Sukhatme, ibid. 54, 2184 (1985).
- G. Darboux, C. R. Acad. Sci. (Paris) 94, 1456 (1882).
- P. B. Abraham and H. E. Moses, Phys. Rev. A 22, 1333 (1980). Hereafter, this paper will be referred to as AM.
- Marshall Luban and D. L. Pursey, Phys. Rev. D 33, 431 (1986).
- See, however, B. Baumgartner, H. Grosse, and A. Martin, Nucl. Phys. B vec 254, 528 (1985).
- D. L. Pursey, report, 1985 (unpublished).
- I. M. Gel'fand and B. M. Levitan, Am. Math. Soc. Transl. 1, 253 (1951).
- The AM method, and also Crum's generalization of the Darboux method [M. M. Crum, Quart. J. Math. 6, 121 (1955)] can add or delete arbitrarily many states in a single step, as can a simple extension of the method developed here, but in this paper I shall confine my attention to the simpler case of adding, deleting, or renormalizing a single state.
- V. A. Marchenko, Dokl. Akad. Nauk SSSR 104, 695 (1955).
- R. G. Newton, J. Math. Phys. 21, 493 (1980).