Google Search can expose a search box scoped to your website when it appears as a search result. This search box is powered by Google Search. However, if you wish to power this search box with your own search engine, or if you want search results also to include an associated mobile app, you can do so using structured data embedded on your website.
Search users sometimes enter the brand name or URL of a known site or app, only to do a more detailed search once they reach their destination. For example, users searching for pizza pins on Pinterest would type Pinterest or pinterest.com into Google Search—either from the Google App or from their web browser—then load the site or Android app, and finally search for pizza. The search box provides a quick way for users to search your site or app immediately on the search results page. The search box implements real-time suggestions and other features. (Original articles from Google:https://developers.google.com/search/docs/data-types/sitelinks-searchbox
“query-input”: “required name=search_term_string”