Abstract
Abstract In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.
Highlights
Erwin Schrödinger won the 1933 Nobel prize for physics mainly due to the paper ‘Quantisierung als Eigenwertproblem’ which appeared in 1926 as a series of four articles in the rst of which [32, p. 362] a version of the Schrödinger equation is announced.The article relevant for us is the third article, [33]
In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured
We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries
Summary
Erwin Schrödinger won the 1933 Nobel prize for physics mainly due to the paper ‘Quantisierung als Eigenwertproblem’ which appeared in 1926 as a series of four articles in the rst of which [32, p. 362] a version of the Schrödinger equation is announced. It is a general study of relevant parts of that theory and does not make direct connection to physics. These developments are made with the objective to explain in Part II, titled ‘Applications to the Stark e ect’ (and consisting of §3,§4,§5), the splitting of the spectrum of light emitted in the presence of strong electric elds (Stark e ect).
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