| Summary: | We consider the linear, second-order elliptic, Schrödinger-type differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">L</mi><mo>:</mo><mo>=</mo><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msup><mo>∇</mo><mn>2</mn></msup><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>r</mi><mn>2</mn></msup><mn>2</mn></mfrac></mstyle><mo>.</mo></mrow></semantics></math></inline-formula> Because of its rotational invariance, that is it does not change under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> transformations, the eigenvalue problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="[" close="]"><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msup><mo>∇</mo><mn>2</mn></msup><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>r</mi><mn>2</mn></msup><mn>2</mn></mfrac></mstyle></mfenced><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>λ</mi><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> can be studied more conveniently in spherical polar coordinates. It is already known that the eigenfunctions of the problem depend on three parameters. The so-called <i>accidental degeneracy</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">L</mi></semantics></math></inline-formula> occurs when the eigenvalues of the problem depend on one of such parameters only. We exploited ladder operators to reformulate accidental degeneracy, so as to provide a new way to describe degeneracy in elliptic PDE problems.
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