Clasine Van Winter discussing N-body quantum theory with Israel Sigal (University of Toronto) at the Mittag-Leffler Institute, Djursholm, Sweden, during the 1981-1982 Special Year in Mathematical Physics. Click on the picture for a larger image.

Clasine van Winter died on October 16, 2000 after a long bout with cancer. She retired in January 2000 from a joint Professorship in the Departments of Mathematics and Physics and Astronomy at the University of Kentucky.

Clasine joined the faculty of the University of Kentucky as a full professor in 1968. She received her PhD from the University of Groningen, the Netherlands, in 1957. She held positions at the University of Birmingham, England, the Niels Bohr Institute in Denmark, Indiana University, the Mittag-Leffler Institute in Sweden, and at the Argonne National Laboratory. She was awarded a prestigious University Research Professorship at the University of Kentucky for the academic year 1981-82.

Clasine's research was devoted to the study of the spectral and scattering theory of N-body Schrodinger operators and resonances in quantum systems. She independently proved the result, now referred to as the HVZ Theorem for Hunziker, van Winter, and Zhislin on the location of the bottom of the essential spectrum of N-body operators, in 1964 [2]. In its simplest form, the HVZ theorem states that the continuous spectrum of an atom with N electrons begins at the ground state energy for an atom with N-1 electrons. This result can be extended to other multi-particle systems, but a precise statement is rather technical.

To appreciate its significance, one can go back to a celebrated 1951 paper of T. Kato [1] which is often cited as having proved that the Hamiltonian for a Helium atom has an infinite discrete spectrum. Although Kato did, in essence, prove this result, he could only state the existence of a ''very large number'' of eigenvalues because the HVZ theorem had not yet been proved. In 1960, Zhislin proved the atomic result. In 1964, van Winter, using a completely different method based on the so-called Weinberg-van Winter equations, proved the HVZ theorem for a much larger class of Hamiltonians [3].

Another of her contributions, the Weinberg-van Winter equations, expresses the resolvent of an N-body operator in terms of the resolvents of 2-body subsystems. The Weinberg-van Winter equations were developed independently by van Winter [3] and Steven Weinberg [6] and have played an important role in N-body scattering theory. For additional information, see Section XIII.5 of [2], pp. 120-135 and the historical notes on p. 343.

Many of Clasine's papers are dedicated to the study of the analytic properties of Green's functions and the wave operators for quantum systems. Consequently, Clasine developed enormous expertise in the theory of Hilbert spaces of analytic functions and analytic Fredholm theory. She formulated the N-body problem with analytic interactions on Hilbert spaces of analytic functions, and used this formulation to prove the existence of the meromorphic continuation of matrix elements of the resolvent [4,5]. She identified the isolated poles of the continuation as resonances of the scattering matrix. Clasine studied in detail the wave operators and their continuation for three-body systems with dilation-analytic potentials. Her most recent papers deal with irreversibility and chaos in quantum mechanics.

Clasine was a dedicated and conscientious teacher, always worth listening to in our collective deliberations, and a stalwart exponent of high standards in all aspects of our academic life. She will be greatly missed.

The Department of Mathematics
The Department of Physics and Astronomy

References

1. T. Kato. On the existence of solutions of the helium wave equation. Trans. Amer. Math. Soc. 70, (1951) 212-218.
2. M. Reed, B. Simon. Methods of Modern Mathematical Physics, IV: Analysis of Operators. New York: Academic Press, 1978.
3. C. van Winter. Theory of finite systems of particles. I. The Green function. Mat.-Fys. Skr. Danske Vid. Selsk. 2 1964 no. 8, 60 pp. (1964).
4. C. van Winter. Complex dynamical variables for multiparticle systems with analytic interactions. I. J. Math. Anal. Appl. 47 (1974), 633-670.
5. C. van Winter. Complex dynamical variables for multiparticle systems with analytic interactions. II. J. Math. Anal. Appl. 48 (1974), 368-399.
6. S. Weinberg. Systematic solution of multiparticle scattering problems. Phys. Rev. 135B (1964), 800-803.