Analytical solution of the Schrödinger equation with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> <mml:mn>3</mml:m

We study the Schrödinger equation with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:msup> <a:mi>r</a:mi> <a:mn>3</a:mn> </a:msup> </a:mrow> </a:math> and attractive <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mn>1</b:mn> <b:mo>/</b:mo> <b:msup> <b:mi>r</b:mi> <b:mn>2</b:mn> </b:msup> </b:mrow> </b:math> potentials. Using the quantum defect theory, we obtain analytical solutions for both repulsive and attractive <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mn>1</c:mn> <c:mo>/</c:mo> <c:msup> <c:mi>r</c:mi> <c:mn>3</c:mn> </c:msup> </c:mrow> </c:math> interactions. The obtained discrete-scale-invariant energies and wave functions, validated by excellent agreement with numerical results, provide a natural framework for describing the universality of Efimov states in mixed dimension. Specifically, we consider a three-body system consisting of two heavy particles with large dipole moments confined to a quasi-one-dimensional geometry and resonantly interacting with an unconfined light particle. With the Born-Oppenheimer approximation, this system is effectively reduced to the Schrödinger equation with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mn>1</d:mn> <d:mo>/</d:mo> <d:msup> <d:mi>r</d:mi> <d:mn>3</d:mn> </d:msup> </d:mrow> </d:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mn>1</e:mn> <e:mo>/</e:mo> <e:msup> <e:mi>r</e:mi> <e:mn>2</e:mn> </e:msup> </e:mrow> </e:math> potentials, and manifests the Efimov effect. Our analytical solution suggests that, for repulsive dipole interactions, the three-body parameter of the mixed-dimensional Efimov states is universally set by the dipolar length scale, whereas for attractive interactions it explicitly depends on the short-range phase. We also investigate the effects of finite transverse confinement and find that our analytical results are useful for describing the Efimov states composed of two polar molecules and a light atom.