<p>In this article, we consider the Schrödinger equation for the Grushin operator with a suitable initial condition <i>f</i>. The solution of this equation is formally denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(e^{-isG}f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>s</mi> <mi>G</mi> </mrow> </msup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>. We discuss the analytic extension property of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(e^{-isG}f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>s</mi> <mi>G</mi> </mrow> </msup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> and characterize the image of a subspace of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(\mathbb {R}^{n+1})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(e^{-isG}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>s</mi> <mi>G</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> as a direct sum of two weighted Bergman spaces.</p>